Arrhenius Rate Ratio
How much faster a reaction runs at a new temperature, from the activation energy.
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The engineering
The two-temperature form of Arrhenius's equation: the pre-exponential factor cancels, so activation energy alone sets how steeply rate climbs with temperature. The lab folklore that 'rates double every 10 °C' is exactly true only for Ea ≈ 50 kJ/mol near room temperature — this card's defaults reproduce it (25→35 °C gives k₂/k₁ ≈ 1.92) — and the rule drifts for other energies and temperatures, which the Q₁₀ row makes visible.
Assumptions: a single rate-limiting step with temperature-independent Ea over the span, and no change of mechanism — a catalyst deactivating or a diffusion limit kicking in breaks the extrapolation. The same math runs accelerated-aging tests: electronics stress at 85 °C to predict decades at 25 °C lean on exactly this exponential, and on the same caveats.
Where this math comes from
Jacobus van 't Hoff noticed in 1884 that equilibrium constants follow an exponential in 1/T; Svante Arrhenius made the leap for *rates* in 1889, proposing that only molecules with energy above a barrier can react — the 'activation energy' — years before anyone could say what the barrier physically was.
Henry Eyring's transition-state theory (1935) finally gave the barrier a face: a saddle point on a potential-energy surface. Arrhenius, whose rate law was initially received coolly (his dissertation nearly failed), went on to a Nobel Prize in 1903 — for the ionic-dissociation work, not the equation every process, pharma, and reliability engineer now uses daily.
- 1884Jacobus van 't HoffTemperature dependence of equilibrium — the exponential form.
- 1889Svante ArrheniusActivation energy and the rate equation.
- 1935Henry EyringTransition-state theory explains the barrier.
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