An aluminum alloy reaches peak hardness after 10 hours of aging at 190 °C. If the aging rate follows the Arrhenius form with an activation energy of 120 kJ/mol, which value is closest to the time needed to reach peak hardness at 160 °C? Take R = 8.314 J/(mol·K).

Answer: C. About 87 hours

If the reaction rate is proportional to exp(−Q/RT), the time to reach the same state is t ∝ exp(Q/RT). With T₁ = 463 K and T₂ = 433 K, t₂ = t₁ × exp[(Q/R)(1/T₂ − 1/T₁)] = 10 × exp(14433 × 1.495×10⁻⁴) = 10 × exp(2.16) ≈ 87 hours. About 1.2 hours reverses the sign of the exponent, giving a shorter time even though the temperature was lowered. About 12 hours assumes the time is inversely proportional to the temperature in °C. The roughly ninefold increase in time for a difference of only 30 °C reflects the strong temperature dependence of the Arrhenius form.