The temperature dependence of the rate constant ( k ) for a chemical reaction is given by the equation: k = 15 - 8000 T where T is the temperature in Kelvin. What is the activation energy ( E _ a ) for this reaction? (Given R = 8.314 ~J ~K ⁻¹ ~mol ⁻¹ )

Step-by-step solution

According to the Arrhenius equation: k = A e ^ - E _ a / RT Taking the natural logarithm on both sides gives: k = A - E _ a RT Comparing this with the given equation k = 15 - 8000 T , we get: E _ a R = 8000 E _ a = 8000 R E _ a = 8000 8.314 ~J ~mol ⁻¹ E _ a = 66512 ~J ~mol ⁻¹ = 66.51 ~kJ ~mol ⁻¹ Mistakes in distractors: Option 1 assumes the equation is in base-10 logarithm and incorrectly multiplies by 2.303. Option 2 incorrectly assumes the coefficient of 1/ T is E _ a directly in kJ ~mol ⁻¹ , forgetting to multip