Ex 5] The rate constant for the first-order isomerization of cyclopropane, C3H6

Question

Ex 5] The rate constant for the first-order isomerization of cyclopropane, C3H6 is 6.7 x 10^-4 s^-1 at 500*C. What is the value of 300*C? A=11.6 x 10^15 s^-1, Ea=272 kJ/mol.
Why do they give the first part 6.7 x 10^-4 and temperature in the first part. I didn't use that taking the Arrhenius Equation.
k=2.557 x 10^-10

so for Ex6]
Solve for k(300) for the proceding example only using k(500) =6.7 x 10^-4 s^-1 and Ea=272 kJ/mol. Do not use the value for the pre-exponential factor, --> what is this saying?

Please explain Example 6 in detail. I don't understand the following example. what k(300) and k(500) means. what they mean by the do not use value for the pre-exponenential factor. I also do not understant what they mean by using the preceding example to only use k(500)=6.7 x 10^-4 and Ea=272 .
Please please explain. I was sick and I haven't been to lecutres so I don't even have a clue.

Explanation / Answer

In ex5, they give you those values to do part 6, but as you say with A and Ea you can do K at 300 °C. So, don't bother about that, and put the value you calculated

The Arrhenius equation is as follow:

K = Aexp(-Ea/RT)

The meaning of K(300) and K(500) means the values of the rate constant at those temperature. Now, how are you going to solve for K(300) without using A, well, you need to use the ratio of K(300) and K(500), and you do that like this. Writting the rate constant for both temperatures. LEt's say that K1 is at 300 °C and K2 at 500 °C (Although you can label them inverse and you should get the same result).

K1 = Aexp(-Ea/RT1)

K2 = Aexp(-Ea/RT2)

The ratio K1/K2 would be:

K1/K2 = Aexp(-Ea/RT1) / Aexp(-Ea/RT2) ----> In this step, the value of A is neglected so:

K1/K2 = exp[(-Ea/RT1) - (-Ea/RT2)] ---->Let's take a common factor Ea/R and we have:

K1/K2 = exp[Ea/R (1/T2 - 1/T1)] ---> as K1 is K(300), solve for K1

K1 = K2exp[Ea/R (1/T2 - 1/T1)]

From there, just put the values of K2, Ea (in J/mol) and temperatures (Both at Kelvin) and using R = 8.3144 J/mol K

K1 = 6.7x10-4 exp (272000/8.3144 (1/773 - 1/573)]

K1 = 2.57x10-10 s-1

Hope this helps

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