A first order reaction has an activation enery of 50.0kJ/mol. When a catalyst is added the activation energy of the catalyzed reaction is Ea/2. Calculate the ration of k(uncatalyzed)/k(catalyzed) at 25 degree C.
I looked at this but didn't finish. I was looking at ln k = -Ea/(RT) + ln(A).
To calculate the ratio of k(uncatalyzed)/k(catalyzed), we need to use the Arrhenius equation. The Arrhenius equation describes the temperature dependence of the rate constant, k:
k = Ae^(-Ea/RT)
Where:
- k is the rate constant
- A is the pre-exponential factor (frequency factor)
- Ea is the activation energy
- R is the gas constant (8.314 J/mol·K)
- T is the temperature in Kelvin
First, we need to convert the activation energy from kJ/mol to J/mol by multiplying it by 1000:
Ea = 50.0 kJ/mol = 50,000 J/mol
Given that the activation energy of the catalyzed reaction is Ea/2, we can calculate the new activation energy for the catalyzed reaction:
Ea_catalyzed = (1/2) * Ea = (1/2) * 50,000 J/mol = 25,000 J/mol
Now, we can calculate the ratio of k(uncatalyzed)/k(catalyzed) using the Arrhenius equation at 25 degrees Celsius (which is 298 Kelvin):
k_uncatalyzed = A_uncatalyzed * e^(-Ea/RT)
k_catalyzed = A_catalyzed * e^(-Ea_catalyzed/RT)
Taking the ratio:
(k_uncatalyzed / k_catalyzed) = (A_uncatalyzed / A_catalyzed) * (e^(-Ea/RT) / e^(-Ea_catalyzed/RT))
Since the temperature and gas constant are the same for both cases, we can cancel them out:
(k_uncatalyzed / k_catalyzed) = (A_uncatalyzed / A_catalyzed) * (e^(-Ea) / e^(-Ea_catalyzed))
Finally, we substitute the given values and solve the equation:
(k_uncatalyzed / k_catalyzed) = (A_uncatalyzed / A_catalyzed) * (e^(-50,000) / e^(-25,000))
Remember, A_uncatalyzed and A_catalyzed are the pre-exponential factors of the respective reactions, which are not provided in the question. You will need to have those values to calculate the ratio of k(uncatalyzed)/k(catalyzed) accurately.