Baromus

Meet Baromus, a climber who finds the air thinner the higher she goes.

Air pressure comes from the weight of the air above you. Climb higher and there is less air on top, so the pressure falls. In a simple model where the temperature is the same at all heights, the pressure falls exponentially, dropping by a factor of e for every scale height H.

The scale height is H = RT / (Mg). For air at 288 K, with a molar mass of 0.029 kg/mol, H ≈ 8.314 × 288 / (0.029 × 9.81) ≈ 8417 m, about 8.4 km.

At the top of Mount Everest, 8849 m up, this model gives p ≈ 101.3 kPa × e^(−8849/8417) ≈ 35 kPa, about a third of the pressure at sea level. Real air gets colder with height, so measured values are a little lower, but the model is close.

With each breath at that height, a climber takes in only about a third as much oxygen as at sea level. Aircraft cabins are pressurised for the same reason, and water boils at a lower temperature on high mountains.