The given reaction,
cyclopropane propene (isomerization),
follows first-order kinetics. The half-live was found to be 4.57 min at 796 K and rate constant was found to be 0.0328 s-1 at 850 K. Calculate the activation energy (in kJ/mol) for this reaction. Round your answer to 3 significant figures.
No guarantees, but here goes:
rate constant = C*e^(-Ea/RT)
A half life of 4.57 min = 274 s corresponds to a rate constant of ln2/274 = 2.53*10^-3 s^-1
Ea is the activation energy
0.0328 = C*e^(-Ea/850R)
0.00253 = C*e^(-Ea/796R)
12.96 = e^[-(Ea/R)(1/850 - 1/796)]
Solve for Ea. R = 8.317 J/mole*K
2.562 = (-Ea/R)*(-7.98*10^-5 K^-1)
Ea/R = 3.21*10^4 K
Ea= 2.67*10^5 J/mole
= 267 kJ/mole
To calculate the activation energy for the given reaction, we can use the Arrhenius equation. The Arrhenius equation relates the rate constant of a reaction to the temperature and activation energy. The equation is given as:
k = A * exp(-Ea / (R * T))
where:
k - rate constant
A - pre-exponential factor
Ea - activation energy
R - gas constant (8.314 J/(mol*K))
T - temperature in Kelvin
We are given two sets of data at two different temperatures, 796 K and 850 K. Using this information, we can set up two equations and solve for the activation energy. Let's start by setting up the equation for the first temperature (796 K):
k1 = A * exp(-Ea / (R * T1))
We are also given the half-life, which is related to the rate constant by the equation:
t1/2 = ln(2) / k1
Rearranging this equation to solve for k1, we get:
k1 = ln(2) / t1/2 ⟶ Equation 1
Now let's set up the equation for the second temperature (850 K):
k2 = A * exp(-Ea / (R * T2))
Similarly, we can use the rate constant at this temperature to calculate the half-life:
t1/2 = ln(2) / k2 ⟶ Equation 2
Rearranging Equation 2 to solve for k2, we get:
k2 = ln(2) / t1/2 ⟶ Equation 3
We know that the pre-exponential factor (A) is the same in both equations since it only depends on the reaction and not on the temperature.
Dividing Equation 3 by Equation 1:
k2 / k1 = (ln(2) / t1/2) / (ln(2) / t1/2)
Simplifying, we get:
k2 / k1 = t1/2 / t1/2
Since both sides of the equation are equal to 1, we can write:
k2 / k1 = 1
Now, substitute the given rate constants and solve for Ea:
0.0328 / k1 = 1
k1 = ln(2) / t1/2
0.0328 / (ln(2) / t1/2) = 1
0.0328 * t1/2 / ln(2) = 1
t1/2 = 4.57 min = 4.57 * 60 = 274.2 seconds
0.0328 * 274.2 / ln(2) = 1
Ea / (8.314 * 796) = ln(0.0328) - ln(1)
Ea = (ln(0.0328) - ln(1)) * (8.314 * 796)
Ea ≈ 48.932 kJ/mol
Therefore, the activation energy for this reaction is approximately 48.932 kJ/mol.