For a first-order decomposition reaction, the half-life period decreases from 40 minutes at 27^ C to 20 minutes at 37^ C . What is the activation energy of this reaction? (Given: 2 = 0.301, R = 8.314 J K ⁻¹ mol ⁻¹ )

Step-by-step solution

For a first-order reaction, the rate constant is inversely proportional to the half-life: k = 0.693 t_ 1/2 . At T₁ = 27^ C = 300 K , t_ 1/2 = 40 min , so k₁ = 0.693 40 . At T₂ = 37^ C = 310 K , t_ 1/2 = 20 min , so k₂ = 0.693 20 . Taking the ratio of the rate constants: k₂ k₁ = 40 20 = 2 Using the Arrhenius equation: ( k₂ k₁ ) = E_a 2.303 R ( T₂ - T₁ T₁ T₂ ) Substituting the given values: 2 = E_a 2.303 8.314 ( 310 - 300 300 310 ) 0.301 = E_a 19.147 ( 10 93000 ) E_a = 0.301 19.147 93000 10 E_a = 53598 J mol ⁻¹ = 53.