krate constant · Apre-exponential factor · Eaactivation energy · R= 8.314 J mol⁻¹ K⁻¹ · T in K
Linearized form
ln(k)=ln(A)−REa⋅T1
Slope −Ea/R, intercept ln(A). Foundation of the Arrhenius plot.
Two-point method
Ea=1/T1−1/T2Rln(k2/k1)
A=k1eEa/(RT1)
Exact when the model holds; numerically unstable when ΔT/T<0.1%.
Derived quantities
t1/2=kln2(1st order)
Q10=eEa⋅10/[RT(T+10)]
dTdlnk=RT2Ea
Half-life, temperature coefficient and local thermal sensitivity, evaluated at the calculator's T (or at T_mean in two-point mode).
Activation parameters (Eyring)
ΔH‡=Ea−RT
ΔS‡=R[ln(kBTAh)−1]
ΔG‡=ΔH‡−TΔS‡
Thermodynamic reading of the same fit. Assumes unimolecular condensed-phase reactions; in the gas phase, ΔH‡=Ea−nRT with n = molecularity.
1. Physical meaning
Svante Arrhenius (1889) observed that rate constants grow exponentially with T. His equation combines A (frequency of collisions with correct orientation) and the Boltzmann factor e−Ea/RT — the Maxwell–Boltzmann fraction with energy ≥ Ea. Only collisions satisfying both conditions yield products.
2. Eₐ and A — physical interpretation
Ea is the barrier to the transition state: typical values 40–200 kJ mol⁻¹; below ~10 kJ mol⁻¹ usually indicates diffusion control. A encodes the frequency of effective collisions — for typical unimolecular reactions it falls between 10¹² and 10¹⁶ s⁻¹. Values far outside these ranges warrant explicit mechanistic justification.
3. Derived quantities
t½ = ln 2 / k holds only for first-order kinetics; other orders depend on initial concentration. Q₁₀ (van't Hoff's rule, 1884) quantifies how much a reaction accelerates per +10 K — the lingua franca of biochemistry, physiology and pharmacokinetics. d(ln k)/dT measures the local thermal sensitivity, useful for process control.
4. Activation parameters (Eyring / TST)
Transition State Theory (Eyring, Evans & Polanyi, 1935) re-expresses the Arrhenius fit as thermodynamic quantities of the activated complex: ΔH‡ (activation enthalpy, ≈ Eₐ at room T), ΔS‡ (entropy — negative when the transition state is more ordered than the reactants) and ΔG‡ (free energy barrier that controls the rate). The calculator uses ΔH‡=Ea−RT, valid for unimolecular condensed-phase reactions.
5. Apparent negative Eₐ
Three possible origins, in order of likelihood:
①Experimental error:noise combined with a small ΔT yields a spurious slope. Eₐ < −100 kJ mol⁻¹ has no known physical mechanism.
②Pre-equilibrium:kapp=Keqk2; an exothermic Keq falls with T.
③Barrierless recombination (e.g. H + H → H₂): Eₐ ≈ −2 to −8 kJ mol⁻¹.
6. Limitations and numerical considerations
Arrhenius assumes A and Ea are T-independent — valid over narrow temperature ranges. Curvature of the Arrhenius plot indicates a change of mechanism, quantum tunnelling (relevant below ~200 K for H transfer) or non-Boltzmann distributions; in those cases use Eyring–Evans–Polanyi. The two-point method is ill-conditioned when ΔT/T<0.1%: the calculator warns without blocking, to support educational use. It also warns when k leaves the physical range 10⁻³⁰ – 10¹⁷ s⁻¹.