The half-life of a first-order reaction is 40 minutes at 27^ C and 10 minutes at 47^ C . Calculate the activation energy of the reaction. (Given: R = 8.314 ~J ~K ⁻¹ ~mol ⁻¹ , 4 = 0.6021 )

Step-by-step solution

For a first-order reaction, the rate constant k is inversely proportional to the half-life t_ 1/2 : k = 0.693 t_ 1/2 Therefore, the ratio of rate constants is: k₂ k₁ = (t_ 1/2 )₁ (t_ 1/2 )₂ = 40 10 = 4 Convert temperatures to Kelvin: T₁ = 27^ C + 273 = 300 ~K T₂ = 47^ C + 273 = 320 ~K Using the Arrhenius equation: ( k₂ k₁ ) = E_a 2.303 R ( T₂ - T₁ T₁ T₂ ) (4) = E_a 2.303 8.314 ( 320 - 300 300 320 ) 0.6021 = E_a 19.147 ( 20 96000 ) E_a = 0.6021 19.147 96000 20 E_a = 0.6021 19.147 4800 E_a = 55336.6 ~J ~mol ⁻¹ = 55.3