On Friday, we posed the following back-to-school-themed Fermi problem:
Assuming you're not in a big lecture hall and the professor shuts the door at the start of class, how long does it take for you and your classmates to deplete the oxygen enough to feel it?
We promised a surprising answer, and here it is. You decide if our back-of-the-envelope calculations are reasonable.
Let's build our classroom first. It's 16 feet wide and long, and 10 feet tall. In handy metric dimensions, that's:
5 meters by 5 meters by 3 meters, or 75 cubic meters.
A cubic meter is 1000 liters, so now we've got 75,000 liters of fresh air.
The oxygen content of air is about 21 percent, and at about 17.5 percent you'll run from the room screaming. To get from fresh and breathable to absolutely stifling, take the difference between 21 percent of 75,000 liters and 17.5 percent of 75,000 liters. That gives us 2,625 liters of oxygen to get through.
How much oxygen does a human consume? It was tough finding a reliable source, but this press release about the 2006 installation of a new oxygen generation system on the International Space Station provides a clue:
During normal operations, it will provide 12 pounds daily; enough to support six crew members.
Aha! So one person needs about 2lb of oxygen a day, or .9 kg. But how many liters is that? Oxygen has a molar mass of 16 grams, so oxygen gas, or O2, has a mass of 32 grams per mole. One mole of gas at standard pressure and temperature takes up 22.4 liters. Now, as my high-school chemistry would say, it's time to hop on the mole-train:
.9 kg x (1000 g/1 kg) x (1 mole O2/32 g O2) x (22.4 L/1 mole O2)
This gives us a daily oxygen intake of 630 liters per person. Let's get a more reasonable rate:
(630 L/day) x (1 day/24 hours) x (1 hour/60 mins)
Now we have the serviceable rate of oxygen consumption of .4375 liters per minute. We're almost there.
Now populate the classroom with 34 students and 1 teacher. The 35 occupants consume 15.3125 liters per minute. Now for the final calculation:
2625 L x (1 minute/ 15.3125 L)
It will take about 171 minutes, or 2 hours and 51 minutes for the room to become unbearably stifling. You can image that you'd start to feel pretty uncomfortable about an hour and a half into the lecture—a good argument for shorter classes.