Arrhenius Equation and Activation Energy

Leave milk on the counter and it sours in a day. Put it in the fridge and it lasts a week. Nothing changed except temperature, yet the rate of spoilage collapsed. This sensitivity of reaction rate to temperature is one of the most useful patterns in chemistry. The Arrhenius equation captures that pattern in a single, testable formula. It says that rate constants grow exponentially with temperature, controlled by a single energy barrier. That barrier is the activation energy, $E_a$ . This article builds the Arrhenius equation from data, derives its integrated and linearized forms, and shows how to extract $E_a$ from an experiment. You will see the equation's origin. Then its graphical test, the Arrhenius plot. Finally, how to read physical meaning into the numbers it produces.

Key takeaways

The Arrhenius equation is $k = A e^{-E_a/RT}$ , linking the rate constant $k$ to temperature $T$ .
$E_a$ is the minimum energy molecules need, relative to their average energy, to react.
$A$ , the pre-exponential factor, reflects collision frequency and orientation requirements.
A plot of $\ln k$ against $1/T$ gives a straight line of slope $-E_a/R$ .
Two rate constants at two temperatures let you solve for $E_a$ without knowing $A$ .
Key idea: the exponential term $e^{-E_a/RT}$ is the fraction of molecules with enough energy to react, so rate constants are exquisitely sensitive to both temperature and barrier height.

The core insight

Reaction rates depend on temperature almost entirely through one exponential factor, $e^{-E_a/RT}$ , which measures the fraction of molecular collisions energetic enough to cross a barrier of height $E_a$ ; the Arrhenius equation isolates this factor so that $E_a$ and the collision-related prefactor $A$ can be measured separately from a simple linear plot.

The physical picture

In 1889, Svante Arrhenius noticed that rate constants for many reactions did not increase linearly with temperature. They increased exponentially, and the curvature was the same shape as the Boltzmann distribution of molecular energies. His insight was that only molecules with energy above some threshold, $E_a$ , can react when they collide. Raising the temperature does not speed up every collision equally. It disproportionately increases the number of collisions energetic enough to clear the barrier. Picture a reaction as a ball needing to cross a hill separating reactants from products. The height of the hill is $E_a$ . At any temperature, molecular kinetic energies are spread out according to a Boltzmann-like distribution: most molecules have modest energy, and a shrinking tail extends to high energy. Only the tail beyond $E_a$ contributes to reaction. Because that tail grows exponentially with $T$ , small temperature increases can dramatically increase the reaction rate — the familiar rule of thumb that many reaction rates roughly double for every 10 K rise near room temperature. The pre-exponential factor $A$ packages everything else: how often molecules collide, and how often those collisions are correctly oriented for reaction. In simple collision theory, $A$ is related to collision frequency and a steric factor; in the more refined transition-state picture (a later topic), $A$ connects to the entropy of forming an activated complex. For now, treat $A$ as an empirical constant with the same units as $k$ .

Figure

The energy barrier picture: reactants must clear height

E_a

to reach products

Figure

Arrhenius plot: ln k versus 1/T gives slope minus Ea over R

The core relations

k = A\, e^{-E_a/RT}

where: $k$ is the rate constant, $A$ the pre-exponential (frequency) factor, $E_a$ the activation energy (J mol⁻¹), $R = 8.314$ J K⁻¹ mol⁻¹, and $T$ the absolute temperature.

\ln k = \ln A - \frac{E_a}{RT}

where: this is the linearized form; plotting $\ln k$ against $1/T$ gives slope $-E_a/R$ and intercept $\ln A$ .

\ln\!\left(\frac{k_2}{k_1}\right) = -\frac{E_a}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right)

where: the two-point form, letting $E_a$ be found from rate constants $k_1, k_2$ at temperatures $T_1, T_2$ without knowing $A$ .

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

where: the differential (van 't Hoff-style) definition of $E_a$ , showing it can be extracted from the local slope of $\ln k$ versus $T$ even if $A$ and $E_a$ are themselves weakly $T$ -dependent.

Derivation AFrom the Boltzmann tail to the Arrhenius law

Goal. Show why an exponential temperature dependence, governed by a single energy parameter $E_a$ , is the natural consequence of a reaction requiring a minimum collision energy.

Step 1: Energy distribution of colliding molecules. In a gas or solution at temperature $T$ , the fraction of molecular encounters possessing translational energy along the reaction coordinate exceeding a threshold $E_a$ is given, from Boltzmann statistics, by a term proportional to $e^{-E_a/RT}$ . This is the same exponential that governs barometric pressure decay with altitude and Boltzmann population ratios generally: high-energy states are exponentially rarer.

Step 2: Total collision frequency sets the prefactor. Not every collision is reactive even if energetic enough; molecules must also be suitably oriented and close enough for bonds to reorganize. Collision theory writes the rate constant as a collision frequency factor $z$ (which itself depends weakly on $T$ , typically as $T^{1/2}$ ) multiplied by a steric factor $p$ accounting for orientation, multiplied by the energetic fraction:

k = p\, z\, e^{-E_a/RT}.

Arrhenius's original empirical proposal lumps $pz$ into one nearly temperature-independent constant $A$ , since the exponential term varies far more steeply with $T$ than any power-law prefactor.

Step 3: Empirical form and its logarithmic linearization. Writing $A$ for the combined prefactor gives the Arrhenius equation,

k = A e^{-E_a/RT}.

Taking natural logs of both sides converts the exponential relationship into a linear one:

\ln k = \ln A - \frac{E_a}{RT}.

This is now a straight line in variables $\ln k$ and $1/T$ , with slope $-E_a/R$ .

Step 4: Eliminating $A$ between two temperatures. Suppose $k_1$ is measured at $T_1$ and $k_2$ at $T_2$ . From Step 3,

\ln k_1 = \ln A - \frac{E_a}{RT_1}, \qquad \ln k_2 = \ln A - \frac{E_a}{RT_2}.

Subtracting the first equation from the second cancels $\ln A$ entirely:

\ln k_2 - \ln k_1 = -\frac{E_a}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right).

Rearranging the left side as a single logarithm gives the boxed two-point Arrhenius relation:

\boxed{\ln\!\left(\frac{k_2}{k_1}\right) = \frac{E_a}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)}

This form lets a chemist determine $E_a$ from just two rate-constant measurements, without ever needing to isolate $A$ .

Worked examples

Example 1 — Finding $E_a$ and $A$ from two rate constants (foundational). A first-order gas-phase decomposition has rate constant $k_1 = 3.46 \times 10^{-5}\ \text{s}^{-1}$ at $T_1 = 298.0$ K and $k_2 = 6.14 \times 10^{-4}\ \text{s}^{-1}$ at $T_2 = 320.0$ K. Find $E_a$ and $A$ . Step 1. Compute the ratio of rate constants:

\frac{k_2}{k_1} = \frac{6.14\times10^{-4}}{3.46\times10^{-5}} = 17.75.

\ln(17.75) = 2.877.

Step 2. Compute the temperature term:

\frac{1}{T_1} - \frac{1}{T_2} = \frac{1}{298.0} - \frac{1}{320.0} = 3.3557\times10^{-3} - 3.1250\times10^{-3} = 2.307\times10^{-4}\ \text{K}^{-1}.

Step 3. Solve the two-point relation for $E_a$ :

E_a = \frac{R \ln(k_2/k_1)}{\left(1/T_1 - 1/T_2\right)} = \frac{8.314 \times 2.877}{2.307\times10^{-4}}.

E_a = \frac{23.92}{2.307\times10^{-4}} = 1.037\times10^{5}\ \text{J mol}^{-1} = 103.7\ \text{kJ mol}^{-1}.

Step 4. Back out $A$ using either data point, say $T_1 = 298.0$ K:

\ln A = \ln k_1 + \frac{E_a}{RT_1} = \ln(3.46\times10^{-5}) + \frac{1.037\times10^{5}}{8.314 \times 298.0}.

\ln(3.46\times10^{-5}) = -10.271, \qquad \frac{1.037\times10^{5}}{2477.6} = 41.86.

\ln A = -10.271 + 41.86 = 31.59 \ \Rightarrow\ A = e^{31.59} = 5.3\times10^{13}\ \text{s}^{-1}.

Sanity check: $A \approx 5 \times 10^{13}\ \text{s}^{-1}$ is a very typical pre-exponential factor for a unimolecular gas-phase decomposition (comparable to a bond-vibration frequency), and $E_a \approx 104$ kJ mol⁻¹ is a plausible barrier for breaking a moderately strong bond. Both numbers are physically reasonable.

Example 2 — Predicting a rate constant at a new temperature; the "rule of thumb" doubling (applied). A reaction has $E_a = 75.0$ kJ mol⁻¹ and $k = 2.20\times10^{-3}\ \text{L mol}^{-1}\text{s}^{-1}$ at $T_1 = 300.0$ K. (a) Predict $k$ at $T_2 = 310.0$ K. (b) Check whether the common "rate doubles every 10 K" rule applies here. Step 1. Compute the temperature term:

\frac{1}{T_1} - \frac{1}{T_2} = \frac{1}{300.0} - \frac{1}{310.0} = 3.3333\times10^{-3} - 3.2258\times10^{-3} = 1.075\times10^{-4}\ \text{K}^{-1}.

Step 2. Evaluate the right-hand side of the two-point equation:

\ln\!\left(\frac{k_2}{k_1}\right) = \frac{E_a}{R}\left(\frac{1}{T_1}-\frac{1}{T_2}\right) = \frac{7.50\times10^{4}}{8.314}\times 1.075\times10^{-4}.

= 9021 \times 1.075\times10^{-4} = 0.9698.

Step 3. Exponentiate to get the ratio:

\frac{k_2}{k_1} = e^{0.9698} = 2.638.

Step 4. Compute $k_2$ :

k_2 = 2.638 \times 2.20\times10^{-3} = 5.80\times10^{-3}\ \text{L mol}^{-1}\text{s}^{-1}.

Step 5. Compare to the "doubling" rule: the ratio 2.64 is somewhat larger than the "double every 10 K" folklore value of 2, confirming that the rule is only a rough approximation. It works best near room temperature for $E_a$ in the range 50–65 kJ mol⁻¹; here $E_a = 75.0$ kJ mol⁻¹ is higher, so the true increase (a factor of 2.64) exceeds the folk estimate. This makes physical sense: a larger barrier means the exponential term is more sensitive to temperature, not less.

Limiting cases & common traps

Situation	What to watch for
$E_a = 0$	$k = A$ , temperature-independent; occurs for some diffusion-controlled or barrierless radical recombination reactions
Very large $E_a$	Rate is extremely temperature-sensitive; small $T$ errors cause large $k$ errors — control temperature tightly in kinetics experiments
Non-linear Arrhenius plot	Signals a change in mechanism, a pre-equilibrium, or a temperature-dependent $A$ (e.g., $A \propto T^{1/2}$ or $T^n$ ); use the modified Arrhenius form $k = AT^n e^{-E_a/RT}$
Negative apparent $E_a$	Occurs in complex mechanisms (e.g., some radical chain or termolecular reactions) where increasing $T$ shifts a pre-equilibrium unfavorably; does not mean "negative barrier" for the elementary step
Enzyme-catalyzed reactions	Apparent $E_a$ often drops sharply above an optimal temperature due to denaturation, not a true kinetic effect

The chemist behind it

Svante Arrhenius proposed his equation in 1889 while studying how sucrose inversion rates changed with temperature, building on earlier qualitative observations by van 't Hoff. He recognized that the temperature dependence mirrored the Boltzmann-Maxwell distribution of molecular energies, connecting chemical kinetics directly to statistical mechanics — a bridge that was strikingly modern for its time. Arrhenius later won the 1903 Nobel Prize in Chemistry, primarily for his theory of electrolytic dissociation, but the activation-energy equation bearing his name remains his most widely used contribution in everyday laboratory practice.

Origins · 1 minSvante Arrhenius and how this was discovered

Svante Arrhenius

1859–1927 · Swedish chemist who linked reaction rates to molecular energy distributions, founding modern chemical kinetics

Ask the tutor: How does the Arrhenius pre-exponential factor connect to the collision frequency and steric factor in collision theory?

Why is the Arrhenius plot drawn as ln k versus 1/T rather than k versus T

Because $k = Ae^{-E_a/RT}$ is an exponential function of $1/T$ , plotting $k$ directly against $T$ gives a curve that is hard to analyze quantitatively. Taking logarithms turns the relationship into a straight line, $\ln k = \ln A - (E_a/R)(1/T)$ . Straight lines are far easier to fit, and the slope and intercept directly hand you $E_a$ and $A$ without needing nonlinear regression. This linearization trick is standard whenever an underlying law is exponential.

Can activation energy be negative

For a genuine elementary reaction step, no — a negative $E_a$ would imply the rate constant decreases as temperature rises, which contradicts the simple collision picture where more energetic collisions never hurt. However, some observed (apparent) activation energies for complex, multi-step mechanisms are negative, usually because a fast pre-equilibrium becomes less favorable at higher temperature, or because a mechanism switches pathways. These are properties of the overall rate law, not of any single elementary barrier.

How precisely can

E_a

be measured from just two data points

Using only two rate constants is convenient but risky, since any experimental error in either $k$ or $T$ propagates directly into $E_a$ with no way to check for outliers. In practice, chemists measure $k$ at five or more temperatures spanning as wide a range as feasible, then fit $\ln k$ versus $1/T$ by least squares. The scatter around the best-fit line also reveals whether the reaction truly obeys simple Arrhenius behavior or hints at a more complex mechanism.

How does Ea relate to the enthalpy of reaction

They are conceptually different quantities. $E_a$ is a kinetic parameter describing the barrier height for the rate-determining step, while $\Delta H$ is a thermodynamic quantity describing the overall energy difference between reactants and products at equilibrium. A reaction can be highly exothermic (large negative $\Delta H$ ) yet still have a large activation energy and proceed slowly at room temperature — combustion of paper is the classic example, stable until ignited.

What comes after the Arrhenius equation in the study of reaction rates

The Arrhenius equation is empirical: it fits data well but does not explain, from first principles, what $A$ physically represents. Collision theory refines this by modeling $A$ from molecular collision frequencies and a steric factor for orientation. Transition-state (Eyring) theory goes further, replacing $A$ with quantities derived from the entropy and structure of an activated complex. Beyond these, some reactions show curved Arrhenius plots requiring temperature-dependent prefactors, motivating extended treatments of temperature dependence.