Wednesday, April 02, 2008

Thoughts on living in a big invisible dome, part 1

This doesn't have anything to do with Midwestern Hemisphere, as such -- it's more a science nerd's thoughts on living in a big, invisible dome. My first thought: when will they run out of oxygen?

First, some facts. A person consumes roughly 2 moles of oxygen (32 grams) per hour. A square meter of grass produces .04 moles of oxygen per hour. Roughly half of the surface of a subdivision is grass; the rest is paved or built upon. Oxygen makes up 21% of the ideal atmosphere. A cubic meter of normal atmosphere at standard temperature and pressure contains roughly 9.4 moles of oxygen. (For you non-science-oriented folk, you might ask what a "mole" is. It's a number -- roughly, six hundred billion million million molecules.) The variables: R (radius of the dome, in meters); H (number of humans in the dome); T (time since the dome started, in hours).

So here goes the math:

Rate of oxygen consumption: 2H moles per hour. Total consumption: 2HT moles.
Rate of oxygen production: .02 Pi*R^2 moles per hour. Total production: .02 Pi*T*R^2 moles.
If you want consumption to equal production, H=.01 Pi*R^2. This is the ratio of people to dome size at which oxygen consumed equals oxygen produced.

But this isn't likely to happen. Probably you'll either have too many people or not enough people for the size of the dome. At any given time T, the amount of oxygen under the dome is 6.3 Pi*R^3 + .02 Pi * R^2 * T - 2HT. The percentage of oxygen in the atmosphere at any given time is this number, divided by (44.6 (moles of gas per cubic meter) * 2/3 Pi*R^3).

When this number hits 35% oxygen, people get edgy and start killing each other. At 50%, everything is dangerously flammable. When it hits 14%, the old and weak start suffocating; at 8%, even the healthy start keeling over. It's interesting to plug in some size and population numbers and see how the math works out. I should mention that for any real size of dome, the bigger risk is too much oxygen; if the dome is a kilometer across, you'd need a population of 7850 to balance out, which seems like a lot. If the dome is 400 meters wide, you need a population of 1,256. I should also mention, the cumulative effect of people and plants is relatively small compared to the volume of oxygen in the dome to start with; hypothetical dome with a diameter of 800 meters (ideal population 5024), with a population of only 800, after 4 weeks, would only show an increase of oxygen from 21% to 21.3%. You'd have to spend years in the dome before oxygen became an issue....

Next: water and sunlight in the giant invisible dome.


Jeff Mountjoy said...

Before any of you smartass math people mention it, yes: I know the percentage of O2 in the atmosphere is limited by the amount of CO2 for plants to process into oxygen. The partial pressure of nitrogen (78% or so) won't change, so there's a hard upper limit to how much oxygen can be created inside the dome. Still, it's a fun math exercise figuring this out.

Tina said...

Given the individuals in my little midwestern subdivision, I'd be far more concerned about methane emissions. NO NEIGHBORHOOD DOME CHILI COOKOFFS.

To challenge your math: What plant species exist in the dome, and as the oxygen saturation changes, which will start to wane and which will thrive (further altering your equations)?

Lou said...

Love the thread. Looking forward to reading part 2.
FYI: Additional "dome dispatch" monologues are being posted at