Consider the kinetic data given in the table below for the first order reaction:

What is the value of the activation energy?

A_2_--> 2A

Expt. Initial [A2] Temp. Initial Rate
mol/L (°C) [A2]·s–1
1 0.10 100 1.2 x 10–3
2 0.20 200 4.8 x 10–3
3 0.15 ? 4.5 x 10–3

Can't you simply plug in k1 and k2 along with T1 and T2 and solve for activation energy with the Arrhenius equation?

I have calculated K1 and K2 by dividing initial rate to initial concentration [A2], and I solved for Ea but for some reason I don't get the right answer, (which is 10.2KJ)

To calculate the activation energy for a first-order reaction, we can use the Arrhenius equation:

k = A * e^(-Ea/RT)

where:
k is the rate constant
A is the pre-exponential factor
Ea is the activation energy
R is the ideal gas constant (8.314 J/(mol·K))
T is the temperature in Kelvin

To find the activation energy, we need to compare two sets of data with different temperatures and calculate the rate constants for each temperature. We can then use the Arrhenius equation to solve for the activation energy.

Let's consider experiments 1 and 2:

Experiment 1:
Initial [A2] = 0.10 mol/L
Temperature = 100°C = 373 K
Initial Rate = 1.2 x 10^(-3) mol/(L·s)

Experiment 2:
Initial [A2] = 0.20 mol/L
Temperature = 200°C = 473 K
Initial Rate = 4.8 x 10^(-3) mol/(L·s)

Using the Arrhenius equation, we can calculate the rate constants (k) for each experiment:

k1 = (1.2 x 10^(-3)) / (0.10) = 1.2 x 10^(-2) L/(mol·s)
k2 = (4.8 x 10^(-3)) / (0.20) = 2.4 x 10^(-2) L/(mol·s)

Now, we can use the rate constants and temperatures in the Arrhenius equation to find the activation energy (Ea):

k1 = A * e^(-Ea/(8.314 * 373))
k2 = A * e^(-Ea/(8.314 * 473))

Dividing both equations:

k2/k1 = e^(-Ea/(8.314 * 473)) / e^(-Ea/(8.314 * 373))

Simplifying:

k2/k1 = e^(-Ea/(8.314 * 473 + Ea/(8.314 * 373)))

Taking the natural logarithm (ln) of both sides:

ln(k2/k1) = -Ea/(8.314) * (1/473 - 1/373)

Now we can substitute the values of k1, k2, and solve for Ea:

ln(2.4 x 10^(-2) / 1.2 x 10^(-2)) = -Ea/(8.314) * (1/473 - 1/373)

ln(2) = -Ea/(8.314) * (1/473 - 1/373)

solving for Ea:

Ea = -8.314 * (1/373 - 1/473) * ln(2)

After calculating this expression, the value of Ea for the first order reaction can be determined.

Please note that in the given table, the temperature for experiment 3 is missing. Without this value, it is not possible to directly calculate the activation energy for the first order reaction using the given data.