A first-order isomerization reaction has a half-life of 69.3 ms . If the frequency factor is 10¹³ s ⁻¹ and the activation energy is 57.44 kJ mol ⁻¹ , the temperature at which the reaction occurs is _________ K (nearest integer). [Given: R = 8.314 J K ⁻¹ mol ⁻¹ , (10) = 2.303 ]

Step-by-step solution

For a first-order reaction, the rate constant k is given by: k = 0.693 t_ 1/2 Given t_ 1/2 = 69.3 ms = 69.3 10⁻³ s = 0.0693 s . k = 0.693 0.0693 = 10 s ⁻¹ According to the Arrhenius equation: k = A e^ -E_a/RT Substituting the given values: 10 = 10¹³ e^ -E_a/RT 10 10¹³ = e^ -E_a/RT 10⁻¹² = e^ -E_a/RT Taking the natural logarithm on both sides: -12 (10) = - E_a RT Rearranging for T : T = E_a 12 R (10) Substituting the values ( E_a = 57.44 kJ mol ⁻¹ = 57440 J mol ⁻¹ ): T = 57440 12 8.314 2.303 T = 57440 229.76 250 K A