Chemistry · General chemistry II · Concept
Activation energy from Arrhenius plots
Use rate constants measured at several temperatures to find the activation energy Ea and the frequency factor A from an Arrhenius plot, or Ea from two temperatures with the two-point form.
What the Arrhenius equation says
Rate constants grow with temperature because more collisions carry enough energy to cross the barrier. The Arrhenius equation splits k into two factors. The frequency factor A counts how often reactants approach the barrier with a suitable orientation, and has the same units as k. The exponential factor e^(−Ea/RT) is the fraction of those approaches with enough energy. It is tiny when Ea is many times RT, which is why a modest temperature rise can multiply k several times.
Make it a straight line
Taking the natural logarithm turns the equation into y = mx + b, with y = ln k and x = 1/T. Plot ln k against 1/T: the slope is −Ea/R and the intercept is ln A. Multiplying the slope by −R gives Ea; raising e to the intercept gives A.
| Plot feature | Equals | So |
|---|---|---|
| x-axis | 1/T (K⁻¹) | Temperatures must be in kelvin |
| y-axis | ln k | Keep one set of k units |
| Slope | −Ea/R | Ea = −slope × 8.314 J/(mol·K) |
| Intercept | ln A | A = e^intercept, in the units of k |
Two temperatures: the two-point form
With only two rate constants, write the Arrhenius equation at each temperature and subtract. ln A cancels, leaving an equation you can solve for Ea, or for a rate constant at a new temperature once Ea is known. Two points always lie on a line, so this gives no check on whether a single Ea really fits.
Units, precision and what the fit means
Use R = 8.314 J/(mol·K) and temperatures in kelvin, then convert Ea to kJ/mol. A is an extrapolation to 1/T = 0, far outside any measured range, so it is much less precise than Ea: report it to one or two significant figures. A straight Arrhenius plot supports one roughly constant Ea over that temperature range. A curved one suggests the mechanism or rate-limiting step changes with temperature.
Why A carries the units of k
The exponential factor is dimensionless, so A must have the same units as k. For a first-order reaction that is s⁻¹; for a second-order reaction, M⁻¹ s⁻¹. In the collision model A = pz: an orientation factor p times a collision frequency z.
Common mistakes
- Plotting k instead of ln k, or T instead of 1/T.
- Using temperatures in °C.
- Reporting the slope itself as Ea, instead of −R × slope.
- Mixing R = 8.314 J/(mol·K) with Ea in kJ/mol without converting.
- Quoting A to as many figures as Ea.
- Assuming a 10 K rise always doubles k: the factor depends on Ea and T.
Key terms
- Arrhenius equation
- k = A·e^(−Ea/RT): how a rate constant grows with temperature. A plot of ln k against 1/T is a line with slope −Ea/R, which gives the activation energy.
- Frequency factor
- The factor A in the Arrhenius equation k = A·e^(−Ea/RT): how often reactant molecules collide with the right orientation.
- Exponential factor
- The term e^(−Ea/RT) in the Arrhenius equation: the fraction of collisions with enough energy to get over the activation barrier. It grows as the temperature rises.
- Rate law
- An equation linking reaction rate to reactant concentrations, such as rate = k[A]²[B]. The units of the rate constant k depend on the overall order.
Work through an example
A first-order reaction has these rate constants at four temperatures. Find the activation energy and the frequency factor.
Find Ea and A from an Arrhenius plot →Sources and scope
Authored study material. Tool results depend on the stated inputs and model assumptions.
- Tro, Chemistry: A Molecular Approach, 4th ed., §14.5 The Effect of Temperature on Reaction Rate, pp. 642–647 (Arrhenius plots, pp. 644–645; two-point form, p. 646)
- OpenStax Chemistry 2e — Collision theory
Try in the workspace
Open the example inputs, change a value and keep a useful result on your board.
Fit the Arrhenius plot in Kinetics See what Ea does to energetic collisions Open worked example on a board Chemistry formulas: experimental analysisYour existing work stays on this device. Examples open as editable copies.