When heated, cyclopropane is converted to propene. Rate constants for this reaction at 470°C and 510°C are k = 1.10 10-4 s-1 and k = 1.02 10-3 s-1, respectively. Determine the activation energy, Ea, from these data.
To determine the activation energy (Ea) from the given rate constants, we can utilize the Arrhenius equation:
k = Ae^(-Ea/RT)
where:
- k is the rate constant
- A is the pre-exponential factor
- Ea is the activation energy
- R is the gas constant
- T is the temperature in Kelvin
We have two sets of rate constants (k1 and k2) for two different temperatures (T1 and T2). By rearranging the Arrhenius equation, we can compare the two rate constants:
k1 = A e^(-Ea/RT1)
k2 = A e^(-Ea/RT2)
Dividing one equation by the other:
k2/k1 = (A e^(-Ea/RT2)) / (A e^(-Ea/RT1))
The pre-exponential factor (A) is the same for both equations and cancels out:
k2/k1 = e^(-Ea/RT2) / e^(-Ea/RT1)
Taking the natural logarithm of both sides:
ln(k2/k1) = -Ea/RT2 + Ea/RT1
Now we can rearrange the equation to isolate the activation energy (Ea):
ln(k2/k1) = (1/RT1 - 1/RT2) * Ea
To calculate the activation energy, substitute the given values:
ln((1.02 x 10^(-3) s^(-1)) / (1.10 x 10^(-4) s^(-1))) = (1/(8.314 J/(mol K) * (470 + 273) K) - 1/(8.314 J/(mol K) * (510 + 273) K)) * Ea
Solving for Ea:
Ea = (ln((1.02 x 10^(-3) s^(-1)) / (1.10 x 10^(-4) s^(-1))) / ((1/(8.314 J/(mol K) * (470 + 273) K) - 1/(8.314 J/(mol K) * (510 + 273) K)))
By plugging in the given values and evaluating the equation, you can determine the value for activation energy (Ea).