If the atmospheric pressure on Mt. Everest is one-third the atmospheric pressure at sea level, what is the partial pressure of oxygen on Everest?
It would still be 20.8% of whatever the total pressure was. See my example below, which results in a partial pressure of slightly more than 7 kPa.
The law of partial pressures ensures us that since oxygen accounts for 20.8% of the molecules in the air, it also accounts for 20.8% of the pressure of the air, regardless of what that total pressure might be.
So, if atmospheric pressure is 102 kPa at sea level (just to choose a convenient value), the partial pressure of the oxygen is
On Everest, the total pressure would be 34 kPa (one-third of 102 kPa), and the partial pressure of the oxygen is