Monday, February 8, 2010

Towards Sustainability in Space

As I was getting on the bus the other day, a fellow passenger stopped me and asked if I was the guy who wrote the “how to estimate anything” book. This was weird for two reasons: (1) I don’t exactly have a following so it’s odd that someone would recognize me and (2) how did this guy I’d never seen before know I took the bus at 7am? Putting the flattered/creepy feeling aside, I struck up a conversation with my would-be stalker. He asked if I knew any way to calculate how many plants you would need to survive in space1.   You could easily calculate how many plants you’d need to produce enough food to survive,2 but my bus mate wanted the answer to a more challenging problem: how many plants would you need to produce enough oxygen to survive?

To estimate an answer, we need to know the rate at which humans consume oxygen and the rate that plants give off oxygen. Previously, I estimated the mass of oxygen consumed each second to be about 5.0×10-4 kg/s. Using the molecular weight of O2 (~32 g/mol), we can compute the number of oxygen molecules lost each second,
rate of O2 consumption = (5.0×10-4 kg/s) / (32 g/mol)
= 0.016 mol/s.

These O2 molecules are lost during respiration when they converted to CO2 via the reaction,

C6H12O6 + 6 O2 → 6 CO2 + 6 H2O.

During photosynthesis, plants take this CO2 from the air and convert it back oxygen,

6 CO2 + 12 H2O + photons → 2 C6H12O6 + 6 O2.

From the chemical equations above, we can see that for each O2 absorbed, a CO2 molecule is emitted. To maintain a constant ratio of O2 to CO2, we need to set the rates of these reactions equal to each other.

To compute the rate at which plants absorb CO2, it’s helpful to have some experience growing plants. The basil in my apartment grows about a foot tall in a couple of months, and, when it’s this size, it weighs about 3 g. From this we can compute the growth rate of the plant:

growth rate = (mass added) / (time)
= (3.0 g) / (2.0 months)
= 5.7×10-10 kg/s

While different species of plants grow at different rates, this seems like a reasonable order of magnitude estimate.

Carbon is the main building block of living things. Different plants have different percentages of carbon in them. If we assume plants have about the same percentage of carbon that glucose has, then they’ll be about 40% carbon by weight3.  Using this estimate, we can calculate the mass of carbon that gets absorbed into plant material each second to be 2.3×10-10 kg/s. Since each of these carbon atoms comes from one CO2 molecule, we can calculate the number of molecules absorbed each second by dividing this by the atomic weight of carbon (~12 g/mol),

rate of CO2 consumption = (2.3×10-10 kg/s) / (12 g/mol)
= 1.9×10-8 mol/s per plant.

It should be noted that this result is the number of molecules absorbed per plant. Again, we need the rate of CO2 consumption to match the rate of O2 consumption. To calculate the number of plants we would need for this to occur, we can divide the CO2 consumption rate into the O2 consumption rate,

# of plants needed = (O2 consumption rate) / (CO2 consumption rate)
= (0.016 mol/s) / (1.9×10-8 mol/s per plant)
= 8.4×105 plants

To sustain breathable air for one person, we would about 840,000 plants. If each plant were to require one square foot of space, you would need about 20 football fields of space to grow them.

[1] Desiree, if more people start stopping me on the street and asking me to calculate stuff on the spot, I’m totally blaming you for it.
[2] This has presumably been solved by Gary Larson, Warner Bros, and a host of other cartoonists with the result being one coconut tree.
[3] At least one reference suggests this estimate is pretty close.

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