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.