In this problem, we are given information about temperature, and information about the rate of reactions. We are told that the reaction is 4 times faster at 25 degrees Celsius, than it is at 0 degrees Celsius. And from this information, we're going to be able to determine the activation energy. The Arrhenius equation has the form, natural log of k2/k1 = negative E sub a/R (1/T2- 1/T1). We're not given rate constants but we do know that since it's a first-order reaction, that the rate at the lower temperature, I'll call it 1 is equal to the rate constant 1 times the concentration of A. And if we were to elevate the temperature, the rate would change and we would have a different rate constant. Now, assuming that the reactions are taking place with the same concentrations, the change in rate has gotta be due solely to the change in the K. Now, if rate 2 as the higher temperature, is 4 times faster then rate 1, It is due to, and we can replace the rate 2 with the K2[A] and the rate 1 with K1[A]. What this tells me is that K2, And K1, all I've done is divided both sides by this portion, those would cancel. And K2/K1 = 4. All of this is to say is if the rate is 4 times faster, then the K's are 4 times larger. And we can replace K2/K1 with 4. E sub a is what we're trying to determine. R, we're going to use the joule one, joule per mole Kelvin. And we will be sure that we use the temperatures in Kelvin. So, the K2 is the higher temperature, that would be 298 Kelvin. The K1 is the lower temperature, it's 273 Kelvin. So the natural log of 4 (8.314), Divided by this portion, which is equal to 3.07 x 10 to the minus 4. And actually this portion is negative, and when we change the sign it become positive, so it's a positive 3.07 x 10 to the minus 4. . The units will be 1 over Kelvin. That will equal the activation energy. This gives me a value of 3.75 x 10 to the 4th. The units will be joules per mole because the Kelvin has cancelled. That's the activation energy in joules per mole, and more commonly, activation energy is reported in kilojoules per mole. We divide by 1,000, we get 37.5 kilojoules per mole.
