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## Q. 13.7

Using an Arrhenius Plot to Determine Kinetic Parameters

The decomposition of ozone is important to many atmospheric reactions.

O$_{3}(g) → O_{2}$(g) + O(g)

A study of the kinetics of the reaction resulted in the following data:

 Temperature (K) Rate Constant (M$^{-1} . s^{-1}$) Temperature (K) Rate Constant (M$^{-1} . s^{-1}$) 600 3.37×10³ 1300 7.83×10$^{7}$ 700 4.85×10$^{4}$ 1400 1.45×10$^{8}$ 800 3.58×10$^{5}$ 1500 2.46×10$^{8}$ 900 1.70×10$^{6}$ 1600 3.93×10$^{8}$ 1000 5.90×10$^{6}$ 1700 5.93×10$^{8}$ 1100 1.63×10$^{7}$ 1800 8.55×10$^{8}$ 1200 3.81×10$^{7}$ 1900 1.19×10$^{9}$

Determine the value of the frequency factor and activation energy for the reaction.

## Verified Solution

To determine the frequency factor and activation energy, prepare a graph of the natural log of the rate constant (ln k) versus the inverse of the temperature (1/T).

The plot is linear, as expected for Arrhenius behavior. The best fitting line has a slope of -1.12 × 10$^{4}$ K and a y-intercept of 26.8. Calculate the activation energy from the slope by setting the slope equal to -E$_{a}$/R and solving for E$_{a}$:
-1.12 × 10$^{4}K = \frac{-E_{a}}{R}$
E$_{a} = 1.12 \times 10^{4} K (8.314 \frac{J}{mol . K})$
= 9.31 ×10$^{4}$ J/mol
= 93.1 kJ/mol
Calculate the frequency factor (A) by setting the intercept equal to ln A.
26.8 = ln A
A = e$^{26.8}$
= 4.36 × 10$^{11}$
Since the rate constants are measured in units of M$^{-1} . s^{-1}$, the frequency factor has the same units. Consequently, we can conclude that the reaction has an activation energy of 93.1 kJ/mol and a frequency factor of 4.36 × 10$^{11} M^{-1} . s^{-1}$.