Activation Energy Calculator

The Arrhenius equation in one line

The Arrhenius equation says that a rate constant k rises with temperature T as k = A x exp(-Ea / (R x T)). Ea is the activation energy in joules per mole, R is the molar gas constant, and A is the pre-exponential factor, also called the frequency factor, which collects collision frequency and orientation effects. The exponential term is the fraction of collisions that carry at least Ea of energy, so raising T raises that fraction and the rate climbs.

The form the calculator uses most is the two point relation. Take the equation at two temperatures, divide one by the other so A cancels, and take logs:

ln(k2 / k1) = (Ea / R) x (1/T1 - 1/T2)

Solve for Ea and you get Ea = R x ln(k2 / k1) / (1/T1 - 1/T2). This is the standard two point form taught in general chemistry; the Florida State University teaching pages derive it exactly this way and note that no knowledge of A is needed (chem.fsu.edu, course material). Temperatures must be absolute, in kelvin, or every term is wrong.

The gas constant and the exact CODATA value

Every joule per mole figure here leans on R. Since the 2019 SI revision, R is fixed by definition from the Boltzmann constant k = 1.380649 x 10^-23 J/K and the Avogadro constant N_A = 6.02214076 x 10^23 /mol, and the product is R = 8.31446261815324 J/(mol x K), exact with zero uncertainty. The CODATA recommended value listed by NIST reads 8.314 462 618... J mol^-1 K^-1, exact (NIST CODATA 2022 adjustment, physics.nist.gov). This calculator uses 8.314462618.

Two unit traps sit next to this constant. First, R must match the units of Ea: with R in J/(mol x K), Ea comes out in J/mol, and this page divides by 1000 to show kJ/mol. Second, the per molecule twin of R is the Boltzmann constant, k_B = R / N_A = 1.380649 x 10^-23 J/K, which equals 8.617333262 x 10^-5 eV/K. Reliability engineering models such as the NIST/SEMATECH e-Handbook Arrhenius acceleration model use that form, with the barrier written as Delta H in electronvolts and typical values of 0.3 to 1.5 eV (itl.nist.gov, NIST/SEMATECH e-Handbook, section 8.1.5.1). The calculator reports the eV per molecule equivalent so the two conventions line up.

Worked example 1: 64.2 kJ/mol from two rate constants

Put k1 = 2.0 x 10^-4 s^-1 at T1 = 300.0 K and k2 = 1.0 x 10^-3 s^-1 at T2 = 320.0 K. The ratio k2/k1 = 5, so ln(k2/k1) = 1.609438. The reciprocal difference is 1/300.0 - 1/320.0 = 0.00020833 1/K. Then:

Ea = 8.314462618 x 1.609438 / 0.00020833 = 64,232 J/mol = 64.23 kJ/mol

Enter the same two points in the calculator and it returns 64.23 kJ/mol, 0.666 eV per molecule, a pre-exponential factor of 3.05 x 10^7 s^-1, and a Q10 of 2.29 over the 300 to 310 K half of that span. The Q10 line answers the familiar rule of thumb directly: near room temperature a barrier of roughly 50 to 60 kJ/mol doubles the rate for every 10 K, which is where the rule comes from and why it is approximate.

Worked example 2: predicting a rate constant at a new temperature

The second mode runs the arrow the other way. Enter k1 = 1.5 x 10^-3 s^-1 at 25.0 degrees C, Ea = 75.0 kJ/mol, and a target of 35.0 degrees C. In kelvin that is 298.15 K to 308.15 K, so:

ln(k2 / k1) = (75000 / 8.314462618) x (1/298.15 - 1/308.15)

The bracket is 1.08829 x 10^-4 1/K and Ea / R = 9021.1 K, so ln(k2/k1) = 0.98175 and k2 = 1.5 x 10^-3 x 2.6695 = 4.00 x 10^-3 s^-1. The rate multiplies by 2.67 for a 10 K rise, a Q10 of 2.67, because the barrier of 75 kJ/mol is steeper than the 50 to 60 kJ/mol that generates the classic doubling rule.

Two readings of how big that is: stored food or a drug degrading with a 75 kJ/mol barrier loses that factor of 2.67 for each 10 degrees of storage warmth, and an Arrhenius plot of ln k against 1/T for the same data has slope -Ea / R = -9021 K.

Reading an Arrhenius plot

Take ln of both sides of k = A x exp(-Ea / (R x T)) and the equation becomes a straight line: ln k = ln A - Ea / (R x T). Plot ln k on the y axis against 1/T on the x axis and the points fall on a line with slope -Ea / R and intercept ln A. That plot is the usual way a lab reports activation energy, and the two point form above is just the line evaluated between two of its points.

A factor of 2 error in Ea shows up as a large error when you extrapolate. Extrapolating a fitted line more than about 20 to 30 K beyond the data is where Arrhenius predictions usually go wrong, so the calculator is most trustworthy inside the range your two measurements bracket.

Ea values for common reaction families

Barriers cluster by reaction type, and the table gives the ranges quoted in standard physical chemistry references. Values are apparent Ea for the overall reaction and can shift with solvent, catalyst, or mechanism.

Where a barrier sits in that table tells you what a 10 K rise buys. A diffusion limited step at 15 kJ/mol speeds up by a factor of 1.22 for a 10 K rise near room temperature, while a 200 kJ/mol decomposition step speeds up by a factor of about 13.7 over the same span. That spread is why storage advice, cooking times, and shelf life estimates do not scale with one universal number.

Eyring, Q10, and the eV convention

Two neighboring frameworks get compared to Ea constantly. The Eyring transition state theory writes k = (k_B x T / h) x exp(-Delta G double dagger / (R x T)) and separates the barrier into enthalpy and entropy parts. For unimolecular solution reactions the link is close to Delta H double dagger = Ea - R x T, so at 298.15 K the enthalpy term runs about 2.48 kJ/mol below Ea, and the entropy term Delta S double dagger follows from ln(k x h / (k_B x T)). Transition state theory splits what Arrhenius lumps into A.

The Q10 temperature coefficient is just the ratio k2/k1 with T2 - T1 fixed at 10 K: Q10 = (k2/k1)^(10/(T2-T1)). Most biochemical processes run a Q10 near 2 to 3 at physiological temperature, and a Q10 of 1 marks a temperature independent process. The calculator computes Q10 from whatever interval you enter, so it works for 5 K or 20 K spans as well.

Finally, the electronvolt convention used in reliability engineering and solid state physics reports the barrier per molecule: Ea in eV equals Ea in kJ/mol divided by 96.485. A 64.23 kJ/mol barrier is 0.666 eV, and the NIST handbook acceleration factor between two temperatures uses the same exponential with k_B in eV/K.

Where the model stops

The Arrhenius equation is an empirical fit and it fails in identifiable ways. Below the glass transition or in enzyme systems past an optimum, rate falls with temperature instead of rising, and a Q10 below 1 is the signature of denaturation rather than of a smaller barrier. Barrierless reactions such as radical recombination can show a negative apparent Ea. Mechanism changes bend the Arrhenius plot, so a single two point estimate inherits whatever the mechanism was doing between the two temperatures. And the pre-exponential factor is not always temperature independent; when A varies as a power of T, a ln k versus 1/T plot curves slightly and a two point Ea picks up a bias.

Use the calculator as a interpolation and planning tool. Measure two rate constants at temperatures that bracket the range you care about, check that the Ea you get back is in a sensible range for the reaction family above, and only then extrapolate a few tens of kelvin. For kinetics work that needs enthalpy and entropy separately, run the Eyring analysis in the mode that reports Delta H double dagger and Delta S double dagger.