Question: What are the chemical reactions that happen in the breathing system? 

Answer: Breathing is a mechanical process of exchanging gases between an organism and its surroundings.

In humans, you use lungs, which absorb gases. When you breathe in air, you take in a mixture of gases:

Oxygen: about 21%
Carbon dioxide: about 0.03%
Nitrogen: about 78%
Rare gases: about 0.97%
Water vapour: amount varies

You breathe out:

Oxygen: about 16%
Carbon dioxide: about 4%
Nitrogen: about 78%
Rare gases: about 0.97%
Water vapour: lots

Some of the oxygen you breathe in goes into the blood stream to be used in respiration.
Carbon dioxide is a waste product of respiration and in the lungs it passes out of the blood and is breathed out.

Respiration is a chemical process when glucose or other sugars react with oxygen to produce energy. It produces carbon dioxide and water as waste products. 

The equation for respiration is: