Reaction Rates & Temperature
Why does food keep longer in the fridge, and why does a reaction go so much faster when you heat it? The Arrhenius equation links the rate constant to the temperature through the activation energy. Find the activation energy from the rate at two temperatures, the rate constant at another temperature or the temperature you need — or fit the activation energy to several measurements with an Arrhenius plot.
What Is the Arrhenius Equation?
Molecules only react when they collide with enough energy to get over an energy barrier: the activation energy Ea. The share of collisions with at least that much energy is about e−Ea/RT. It is tiny, but it grows very fast with the temperature — which is why a small rise in temperature can make a reaction several times faster.
The Arrhenius equation k = A e−Ea/RT combines this with the pre-exponential factor A, which counts how often molecules meet in the right orientation. Taking the logarithm gives a straight line: plot ln k against 1/T (an Arrhenius plot) and the slope is −Ea/R. A catalyst lowers Ea, so the reaction is faster at the same temperature.
Rate Constant and Temperature
kWorked Techniques
Measure the rate constant (or the rate, or 1 / the time a reaction takes) at two temperatures, convert both temperatures to kelvin and fill in.
With a known activation energy, one measurement is enough to predict the rate constant at any other temperature.
A straight line in ln k against 1/T. The least-squares slope gives −Ea/R and the intercept ln A. Points that fall off the line point to a measuring error or a change in mechanism.
Sources & Verification
- P. Atkins, J. de Paula & J. Keeler, Atkins’ Physical Chemistry, 11th ed., Oxford University Press, 2018 — Topic 17D (The temperature dependence of reaction rates).
- IUPAC Compendium of Chemical Terminology (Gold Book) — “Arrhenius equation”, “activation energy”. R = 8.314 462 618 J/(mol·K) (CODATA 2018).
Frequently Asked Questions
How do I calculate the activation energy?
Measure the rate constant (or the rate) at two temperatures and use Ea = R·ln(k₂/k₁) / (1/T₁ − 1/T₂), with the temperatures in kelvin. Example: a reaction runs twice as fast at 35 °C as at 25 °C.
1. T₁ = 298.15 K, T₂ = 308.15 K, so 1/T₁ − 1/T₂ = 0.00010884 K⁻¹.
2. ln(k₂/k₁) = ln 2 = 0.6931.
3. Ea = 8.314 × 0.6931 / 0.00010884 ≈ 52 950 J/mol = 52.9 kJ/mol. In the calculator, click Solve next to the activation energy.
How do I find the rate constant at another temperature?
Use k₂ = k₁·exp[(Ea/R)·(1/T₁ − 1/T₂)]. With Ea = 50 kJ/mol and k₁ = 0.0012 at 25 °C, the rate constant at 60 °C is 0.0012 × 8.32 ≈ 0.00999: the reaction runs 8.3 times as fast. The calculator does the conversion to kelvin for you and also solves for the temperature that gives a certain rate.
What is an Arrhenius plot?
A graph of ln k against 1/T. Taking the logarithm of k = A·e^(−Ea/RT) gives ln k = ln A − (Ea/R)·(1/T): a straight line with slope −Ea/R and intercept ln A. Choose Several measurements and enter your temperatures and rate constants; the calculator fits the line by least squares and gives Ea with its standard error, A and R². For the five example measurements, Ea = 78.3 kJ/mol.
Does the rate double for every 10 °C?
Only roughly. That rule of thumb is true for an activation energy of about 53 kJ/mol near room temperature. With a higher activation energy the rate rises faster: at 25 °C, Ea = 50 kJ/mol gives a factor of 1.92 per 10 °C, while Ea = 100 kJ/mol gives about 3.7. The calculator shows the speed-up per 10 °C for your values.
Why does food keep longer in the fridge?
Spoiling is a set of chemical and biological reactions, and they slow down when it is colder. For an activation energy of 55 kJ/mol, going from 25 °C to 4 °C makes a reaction about 5.4 times slower. Try the fridge example in the calculator; real foods have different activation energies, so this is an illustration.
What is the pre-exponential factor A?
The A in k = A·e^(−Ea/RT). It stands for how often molecules collide in the right orientation, and it is the rate constant you would get if every collision had enough energy. It has the same unit as k. The calculator works it out from your values, or from the intercept of the Arrhenius plot.
Why must the temperature be in kelvin?
Because the Arrhenius equation uses 1/T, which only makes sense on an absolute scale that starts at absolute zero. 25 °C is 298.15 K. You can enter temperatures in °C, K or °F: the calculator converts them before it calculates.
How does a catalyst speed up a reaction?
It offers a different path with a lower activation energy. Because Ea sits in an exponent, a small drop has a big effect: lowering Ea from 50 to 40 kJ/mol at 25 °C makes the reaction about 56 times faster, assuming A stays the same (e^(10 000 / (8.314 × 298.15)) ≈ 56).