Why a Small Rise in Temperature Can Double a Reaction's Rate

Dal cooks faster in a pressure cooker, food keeps longer in a fridge, and a match needs a strike before it bursts into flame. These everyday observations come down to how often molecules collide, how hard they hit, and whether something lowers the energy barrier they must climb.

This lesson covers collision theory, the Arrhenius equation and activation energy, and how catalysts change reaction rates.

Collision theory states that molecules react only when they collide with at least the activation energy and in the correct orientation, so the rate depends on how often they collide and what fraction of those collisions are effective.

Key ideas:

- Reacting molecules are treated as hard spheres that must collide
- Collision frequency is the number of collisions per second per unit volume
- Activation energy, $E_a$, is the minimum extra energy colliding molecules need to react
- Proper orientation means molecules must meet the right way round for bonds to rearrange

Rate expression:

$$\text{Rate}=P{Z}_{AB}{e}^{-E_a/RT}$$

where $Z_{AB}$ is the collision frequency, P, the steric factor, accounts for orientation, and $e^{-E_a/RT}$ is the fraction of collisions with enough energy.

Energy profile. Reactants must pass over an energy hill with an unstable activated complex at the top. The height of the hill above the reactants is $E_a$, and the energy difference between reactants and products is $\Delta H$.

Worked reasoning — orientation. In $\mathrm{C}{\mathrm{H}}_3\mathrm{Br}+\mathrm{O}{\mathrm{H}}^{-}\to \mathrm{C}{\mathrm{H}}_3\mathrm{OH}+\mathrm{B}{\mathrm{r}}^{-}$, the hydroxide ion must approach the carbon from the side opposite the bromine; collisions on the bromine side fail however energetic they are.

Why temperature matters. Heating raises the collision frequency only slightly, but it greatly increases the fraction of molecules with energy above $E_a$ — and that second effect dominates.

An everyday example. A matchstick does not ignite by itself because the molecules in its head lack the activation energy; striking it supplies just enough energy to start a self-sustaining reaction.

The substance. Only a tiny fraction of collisions actually produce a reaction — if every collision worked, most reactions would be over almost instantly.

**The Arrhenius equation, $k=A{e}^{-E_a/RT}$, links the rate constant to temperature, and comparing rate constants at two temperatures gives the activation energy.**

$$k=A{e}^{-E_a/RT}\log k=\log A-\frac{E_a}{2.303RT}$$

- A is the frequency factor, related to collision frequency and orientation
- A plot of $\log k$ against $1/T$ is a straight line with slope $-\frac{E_a}{2.303R}$

Two-temperature form:

$$\log \frac{k_2}{k_1}=\frac{E_a}{2.303R}\left(\frac{T_2-T_1}{T_1{T}_2}\right)$$

Worked example 1. A rate constant doubles when the temperature rises from 298 K to 308 K:

$$E_a=\frac{\log 2\times 2.303\times 8.314\times 298\times 308}{308-298}=\frac{0.301\times 2.303\times 8.314\times 298\times 308}{10}=52900{\text{ J mol}}^{-1}$$

So $E_a$ is about 52.9 kJ mol${}^{-1}$.

Worked example 2 — the fraction of effective molecules. For that $E_a$ at 298 K:

$$e^{-E_a/RT}=e^{-52900/(8.314\times 298)}=e^{-21.35}\approx 5.3\times 10^{-10}$$

Only about five collisions in ten billion carry enough energy.

Rule of thumb. Near room temperature, a 10 K rise roughly doubles or triples the rate constant of many reactions.

An everyday example. Dal cooks faster in a pressure cooker because water boils at about 120 °C inside, and the higher temperature raises the rate constants of the cooking reactions sharply.

The substance. A higher activation energy makes a reaction more sensitive to temperature — the steeper the Arrhenius plot, the more the rate changes for each degree.

A catalyst speeds up a reaction by providing an alternative pathway with a lower activation energy and is chemically unchanged at the end, so it alters neither the enthalpy change nor the position of equilibrium.

How a catalyst works:

- It forms an unstable intermediate with the reactants, which breaks down to give products and release the catalyst
- The lower $E_a$ lets a larger fraction of molecules react at the same temperature
- It speeds up the forward and backward reactions equally, so equilibrium is reached sooner but not shifted
- It changes neither $\Delta H$ nor $\Delta G$, and cannot make a non-spontaneous reaction happen

Worked example. If a catalyst lowers $E_a$ from 75 to 50 kJ mol${}^{-1}$ at 298 K, with A unchanged, the rate constant rises by a factor of

$$\frac{k_{\text{cat}}}{k}=e^{(75000-50000)/(8.314\times 298)}=e^{10.09}\approx 2.4\times 10^4$$

Types and examples:

- Homogeneous — catalyst and reactants in the same phase, as in acid-catalysed hydrolysis of esters
- Heterogeneous — catalyst in a different phase, as with iron in the Haber process and vanadium(V) oxide in the contact process
- Enzymes — biological catalysts, such as amylase in saliva
- Promoters increase a catalyst's activity, while poisons reduce it

An everyday example. Chewing a piece of chapati for a while makes it taste slightly sweet, because amylase in saliva catalyses the breakdown of starch into sugars at body temperature.

The substance. A catalyst lowers the barrier; it does not supply energy — reactants and products keep exactly the same energies as before.