## Chapter 13

## 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.

## Step-by-Step

## 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}.