When climbing a high mountain, you get out of breath more easily primarily because there is less oxygen in each cubic meter of air.
As altitude increases, the air pressure decreases, which means that the number of air molecules—including oxygen molecules—per unit volume of air also decreases. Therefore, even though the ratio of oxygen in the air remains roughly the same (about 21%), there is overall less oxygen available for you to breathe. This results in lower oxygen availability for your body, making it harder to breathe effectively and leading to quicker fatigue and breathlessness as your body struggles to get the oxygen it needs.
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