About this experiment
Most reactions speed up sharply when they are warmed, often doubling for every 10 degrees. In 1889 Svante Arrhenius explained why: only molecules with at least a minimum energy, the activation energy, can react, and the fraction that has it grows exponentially with temperature. Measure rate constants at several temperatures, plot ln k against 1/T to get a straight line, and read the activation energy from its slope, the way it is done in a physical chemistry lab.
Learning goals
- Use the Arrhenius equation to calculate a rate constant at any temperature
- Linearise the equation and find Ea and A from the slope and intercept of ln k against 1/T
- Explain the rule of thumb that a rate doubles for every 10 K and when it fails
- Show how a catalyst changes the slope of the Arrhenius plot
- Judge how the temperature range and measurement error limit the accuracy of Ea
How to use it
- 1Drag along the left plot. How does k change between 290 K and 350 K? Is the rise linear?
- 2Look at the right plot. Why are the measured points on a straight line? Compare the fitted Ea with the true value.
- 3Find the activation energy for which the rate exactly doubles between 298 K and 308 K, using the +10 K readout.
- 4Switch on the catalyst. Which plot changes more? Why does the line get flatter but keep the same intercept?
- 5Set the measurement error to 15 % and the range to 10 K. Run several experiments and note the spread of the fitted Ea. Now widen the range to 100 K.
- 6Compare the two-point estimate with the least squares fit over several runs. Which one is more reliable?
The model
Assumptions
- A and Ea do not depend on temperature over the range shown. Collision theory and transition state theory predict a weak temperature dependence of A, which is ignored.
- The reaction is first order, so k has units of s⁻¹ and a half-life that does not depend on concentration.
- The catalyst lowers Ea by a fixed 30 kJ/mol and leaves A unchanged. Real catalysts provide a different mechanism, which can also change A.
- Each measured k carries a random relative error of the chosen size from a seeded generator, so a run can be repeated exactly. The fit is ordinary least squares on ln k.
Try this in class
- Q1A reaction has Ea = 50 kJ/mol. By what factor does k increase from 20 °C to 30 °C?
- Q2k = 3.0 × 10⁻⁴ s⁻¹ at 300 K and 2.4 × 10⁻³ s⁻¹ at 330 K. Calculate Ea.
- Q3An Arrhenius plot has slope −1.2 × 10⁴ K. What is the activation energy?
- Q4Why do biologists say enzyme activity roughly doubles every 10 °C, but only up to about 40 °C?
- Q5Why is a wider temperature range better for finding Ea, even with the same number of measurements?
Frequently asked questions
Why plot ln k against 1/T and not k against T?
Because the Arrhenius equation becomes a straight line in those variables. A straight line is easy to fit and to check by eye, and its slope gives Ea directly. On k against T the curve is exponential and the activation energy is hard to read.
What does the pre-exponential factor A mean?
It is the rate constant the reaction would have if every collision had enough energy, so it combines how often molecules meet with how often they meet in the right orientation. It has the same units as k.
Does every reaction follow the Arrhenius equation?
Most simple reactions do over moderate temperature ranges. Enzyme reactions stop following it once the protein denatures, some reactions with tunnelling or complex mechanisms curve on the plot, and a few, such as some radical recombinations, even slow down when heated.