For a first order reaction, the half-life is 200 min at 300 K and 20 min at 400 K . The activation energy of the reaction is ________ J mol ⁻¹ . (Given : 10 = 2.3 , R = 8.3 J K ⁻¹ mol ⁻¹ )

Step-by-step solution

For a first order reaction, the rate constant k is inversely proportional to the half-life t_ 1/2 since k = 2 t_ 1/2 . Thus, the ratio of rate constants at 400 K and 300 K is: k₄₀₀ k₃₀₀ = t_ 1/2 (300) t_ 1/2 (400) = 200 20 = 10 Using the Arrhenius equation: ( k₄₀₀ k₃₀₀ ) = E_a R ( 1 T₁ - 1 T₂ ) Substitute the given values: (10) = E_a 8.3 ( 1 300 - 1 400 ) 2.3 = E_a 8.3 ( 400 - 300 300 400 ) 2.3 = E_a 8.3 100 120000 2.3 = E_a 8.3 1200 E_a = 2.3 8.3 1200 = 22908 J mol ⁻¹ Answer: 22908