The half-life of a first-order reaction is 100 minutes at 300 K and 10 minutes at 400 K . The activation energy of the reaction is ________ kJ mol ⁻¹ . [Given: 10 = 2.3 , R = 25 3 J K ⁻¹ mol ⁻¹ ]

Step-by-step solution

For a first-order reaction, the rate constant k is related to the half-life t_ 1/2 by: k = 2 t_ 1/2 Let k₁ and k₂ be the rate constants at T₁ = 300 K and T₂ = 400 K respectively. k₂ k₁ = t_ 1/2 at 300 K t_ 1/2 at 400 K = 100 10 = 10 Using the Arrhenius equation: ( k₂ k₁ ) = E_a R ( 1 T₁ - 1 T₂ ) Substituting the values: (10) = E_a 25/3 ( 1 300 - 1 400 ) 2.3 = 3 E_a 25 ( 400 - 300 300 400 ) 2.3 = 3 E_a 25 100 120000 2.3 = 3 E_a 25 1200 = 3 E_a 30000 2.3 = E_a 10000 E_a = 2.3 10000 = 23000 J mol ⁻¹ Converting to kJ m