On molar mass and ideal gas

I recently started reading Andy Weir’s The Martian which is supposed to be the hardest of hard science fiction, written by the son of a particle physicist and scientifically accurate in every possible respect. We’ve heard that story before, so I was not surprised to find the first error (claiming that desiccated stool would be completely free of bacteria) about 13 pages in. Then I got to page 24 and it got bad. Really, really bad. Bad enough that I wouldn’t be surprised if Weir’s physicist father disowns him.

The protagonist, astronaut Mark Watney, is stranded on Mars and believed dead. He has calculated that he has no chance of surviving until a rescue mission arrives (not least because he has no way of informing anyone that he is still alive), but decides to try anyway. He plans to grow food inside the habitat using a mixture of Martian soil, Terran soil that was brought along for experiments, and his own waste. But he needs water:

There isn’t a lot of water here on Mars. […] I’ll have to make it from scratch. […] Take hydrogen. Add oxygen. Burn.

Burning a stoichiometric mixture of hydrogen and oxygen is actually very dangerous, which is not mentioned, but Watney does reflect on the danger of extracting hydrogen from hydrazine, so I’ll let it slide. But let’s see how he plans on obtaining oxygen:

I have a fair bit of O2 reserves, but […] only enough to make 100 liters of water (50 liters of O2 makes 100 liters of molecules that only have one O each). […] That’s where the MAV fuel plant comes in. […] Once I get the fuel plant hooked up to the Hab’s power, it’ll give me half a liter of liquid CO2 per hour, indefinitely. After ten sols it’ll have made 125 liters of CO2, which will make 125 liters of O2 after I feed it to the oxygenator.

Now for hydrogen, from what’s left in the hydrazine-powered descent module’s fuel tanks:

Each molecule of hydrazine has four hydrogen atoms in it. So each liter of hydrazine has enough hydrogen for two liters of water.

The first red flag is that Watney uses units of volume instead of mass, which is inappropriate when calculating quantities for a chemical reaction. Watney is a mechanical engineer and would have been thoroughly trained in the correct use of units, even if chemistry is not really his field. I also doubt he would use the chemical formulas for carbon dioxide, water etc. in daily conversation or in a diary destined for laypeople, but I understand why Watney (or rather Weir) did it: he wants the reader to be able to count H’s and O’s and follow Watney’s calculations. Unfortunately, his calculations are unsound, because you have to add up mass, not counts.

It is not initially clear whether Watney is talking about gases, liquids or solids. Since he will be working in the habitat, close to standard conditions of temperature and pressure, it is not unreasonable to assume that the CO2, O2 and H2 are in gas form and the H2O is liquid. But it seems Watney himself is confused: when he says that the fuel plant will make “125 liters of CO2, which will make 125 liters of O2 in ten sols, he is right… if he is talking about gases, but not if he is talking about liquids (“it’ll give me half a liter of liquid CO2 per hour”).

In reality, 1 l of liquid CO2 at a density of 770 kg·m-3 contains (770 / 44) * 32 = 560 g of oxygen, barely enough for 0.5 l of liquid O2 at a density of 1141 kg·m-3. Since 1 l of water requires (1000 / 18) * 16 = 889 g of oxygen, 1 l of liquid CO2 will only provide enough oxygen for 0.63 l of water.

Meanwhile, 1 l of liquid N2H4 at 1021 kg·m-3 contains (1021 / 32) * 4 = 128 g of hydrogen, which is enough for slightly more than 1 l of water ((1000 / 18) * 2 = 111 g), not the 2 l Watney claims.

It would be different if he was operating exclusively with gases. Assuming the ideal gas law is sufficiently accurate (which depends on temperature, pressure and molecule size), and assuming conditions of temperature and pressure under which carbon dioxide, hydrazine and water are all in gas form, one liter of carbon dioxide and one liter of hydrazine vapor contain enough hydrogen and oxygen for two liters of water vapor (which is not the same as steam) plus one liter of nitrogen and a few grams of solid carbon.

Finally, Watney mentions that some of the reactions he relies on are extremely exothermic, but not that releasing liquid carbon dioxide into the habitat’s atmosphere will dramatically lower the temperature. The exterior temperature is never mentioned, so I cannot comment on the effect of bringing in soil and hydrazine, nor on the state of the hydrazine, which has a melting point of 2 °C and is therefore very likely to be frozen solid.

I’ll keep reading, for the same reason I sometimes watch CSI (but not CSI Miami): the story and characters are sufficiently engaging that I can overlook the bad science, as long as they’re not waving it in my face. The Martian is flying dangerously close to Gap territory, but at least the text flows well and the characters are likable. For now.

2 thoughts on “On molar mass and ideal gas

  1. Just read the book and have seen the film. Am fascinated by the problem fixes.
    In your calcs can you please give a little info on the formulae you used to extract the O2 and make water. ….. Is it possible to light a H and O2 mixture and not get an explosion?

    1. The mass of 1 m³ (1,000 l) of liquid CO₂ is 770 kg, so the mass of 1 l is 770 g. The molar mass of CO₂ is 12 + 16 * 2 = 12 + 32 = 44 g / mol. 770 g / 44 g = 17.5 mol, 17.5 * 32 g = 560 g O₂. Similar calculations for water, which has a density of 1,000 g / l (or close enough as makes no difference) and a molar mass of 2 + 16 = 18 g / mol, and hydrazine, which has a density of 1,021 g / l and a molar mass of 14 * 2 + 4 = 32 g / mol.

      How safe it is to burn hydrogen, or any other gas, depends on the homogeneity of the mixture and the exact proportion of oxygen to hydrogen. The thinner the gas (and hydrogen is the thinnest combustible gas of all) the harder to control.

      Keywords for Wikipedia: ideal gas law, molar mass, standard conditions for temperature and pressure, stoichiometry.

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