Milk keeps for a week in the fridge but sours in a day on the counter. Bread browns faster in a hotter oven, and a glow stick glows longer when you put it in the freezer. It is the same chemistry every time — what changes is only the temperature. Heating something does not change what can react; it changes how often the molecules slam together hard enough to actually react.
Every reaction has an energy "hill" the molecules must climb before reactants can turn into products. The height of that hill is the activation energy $E_a$. Only molecules carrying at least that much energy make it over. Warm the mixture and a much larger share of molecules carry that much energy, so the reaction speeds up. We measure that speed with the rate constant $k$, and it obeys one compact rule:
Here $A$ counts how often molecules collide while lined up correctly, $R = 8.314$ J·mol⁻¹·K⁻¹, and $T$ is the absolute temperature. The whole story lives in the exponent. As a worked number, take a reaction with $E_a = 50$ kJ/mol and warm it from 300 K to 310 K: the fraction of molecules above the barrier, $e^{-E_a/RT}$, roughly doubles — so the rate roughly doubles for just a 10° rise.
That factor $e^{-E_a/RT}$ is the Boltzmann fraction — the slice of the energy distribution sitting above the barrier. Because it is exponential, "doubling per 10°" is only a rule of thumb that holds near $E_a \approx 50$ kJ/mol; a stiffer barrier such as $E_a = 103$ kJ/mol makes the rate roughly quadruple per 10°. Taking the natural log straightens the curve into a line, $\ln k = \ln A - \frac{E_a}{R}\cdot\frac{1}{T}$, so plotting $\ln k$ against $1/T$ gives a slope of $-E_a/R$ — exactly the "ln k vs 1/T" graph below. The sliders map straight onto the symbols: $E_a$ sets the hill's height (and that slope), $T$ slides you along the line, $\log_{10}A$ shifts the whole line up or down, and the catalyst toggle opens a lower pass without moving $\Delta H$.
Set $E_a$ low (about 30) and then high (about 200) at a fixed temperature, and watch the $k$ readout swing by many orders of magnitude. Next hold $E_a$ fixed and drag $T$ from 300 to 600 K while the "Boltzmann Distribution" graph is open — see the energetic tail past $E_a$ swell as you heat. Finally turn on the catalyst path: the barrier drops and $k$ jumps, yet the products sit at the same height, proof that a catalyst speeds the trip without changing the destination.
Named after Svante Arrhenius (1889), this equation relates the rate constant $k$ to temperature $T$, activation energy $E_a$, and a pre-exponential factor $A$.
| Symbol | Meaning | Unit |
|---|---|---|
| $k$ | Rate constant | s⁻¹ (1st order) or L·mol⁻¹·s⁻¹ (2nd order) |
| $A$ | Pre-exponential (frequency) factor | same as $k$ |
| $E_a$ | Activation energy — minimum kinetic energy for reaction | J·mol⁻¹ (or kJ·mol⁻¹) |
| $R$ | Universal gas constant | 8.314 J·mol⁻¹·K⁻¹ |
| $T$ | Absolute temperature | K |
| $f$ | Boltzmann fraction: $e^{-E_a/RT}$ | dimensionless |
| Slider / Control | Equation Symbol | Effect |
|---|---|---|
| Activation Energy slider | $E_a$ (kJ/mol) | Changes slope of Arrhenius plot, exponential change in $k$ |
| Temperature slider | $T$ (K) | Moves position on Arrhenius plot, shifts Boltzmann distribution |
| Pre-exp factor slider | $\log_{10} A$ | Shifts entire Arrhenius line up/down, changes $k$ at all $T$ |
| ΔH reaction slider | $\Delta H_{rxn}$ | Sets reverse activation energy: $E_{a,\text{rev}} = E_a - \Delta H_{rxn}$ |
| Catalyst toggle | $E_a \to E_a'$ | Lowers $E_a$ by ~40 kJ/mol, dramatically increases $k$ |