I understand that after Ozone is broken down from the energy radiating into the atmosphere, O2 and atomic oxygen are produced. The O2 then eventually repairs with another atomic oxygen to form more Ozone. However, the presence of Chlorofluorocarbons apparently destroys the Ozone layer. When I looked at the reaction of the decomposition of Chlorofluorocarbons, I found that it was CF2Cl2 <--> CF2Cl + Cl, the atomic Chlorine pairs with an Ozone molecule, creating ClO3. This breaks down further into CLO and O2.
My question is, even if the chlorofluorocarbons are present in the atmosphere, it still provides O2 for more Ozone to be created. So, why is it a problem?
The problem is that the addition of CFCs to the atmosphere leads to a net depletion of ozone. Your equations were correct:
Cl + O3 <--> ClO3,
and ClO3 <--> ClO + O2
CFCs essentially add Cl to the atmosphere, which then interact with ozone molecules. Although oxygen is a bi-product of the reactions (listed above), the process doesn't end there because chlorine monoxide (ClO) can then react with another ozone molecule and continue destroying ozone molecules.
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