How many mole of oxygen gas is there in the atmosphere?

https://www.newscientist.com/article/mg15020308-500-the-last-word/ says the atmosphere weighs 5 × 10^21 grams. The molar weight of oxygen gas is 32 gram per mole. Based on that estimate for the weight of atmosphere, the total quantity of O2 in atmosphere is around 5*10^21/32 = 1.5625e+20 mole.

It is easier to calculate:

1. The pressure of air on the ocean level is $approx 100 rm {kPa}$.
2. Thus, its weight is $approx$ 100000N over $1 rm m^2$.
3. Considering that the overwhelming majority of the atmosphere is above some km over the surface of the Earth, we can neglige the decrease of the gravity with height.

Now calculate the mass of the air above a square meter (ten tons). And multiple it with the surface of the Earth (in square meters). Then, divide it with the mean molar mass of air.