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A Pail of Air is a classic science fiction short story by Fritz Leiber from 1951 describing how a family survives on an Earth that has left its orbit around the sun, using desparate low-tech methods: a shelter isolated by many layers of blankets and a coal fire to thaw frozen oxygen gathered in the eponymous pails.

I am curious how realistic this scenario is from a basic chemistry and biology perspective. I think not for a number of reasons, but don't know enough to quantify these factors:

  • Would a coal fire produce enough energy to thaw the oxygen the fire itself uses?
  • Would the resulting atmosphere (at sufficient pressure for the oxygen to support human life) contain toxic levels of carbon dioxide?
  • ...or carbon monoxide?
  • How does oxygen at lower pressures affect flammability? Wouldn't it all go up in a blaze at the drop of a hat?
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  • $\begingroup$ Frozen oxygen? Oxygen turns to a gas at about the temperature that liquid nitrogen does, so a coal fire would melt it for sure. However, liquid oxygen is highly flammable and highly, highly explosive, so what I see happening with melting solid oxygen with a coal fire is a massive explosion. $\endgroup$ – Canageek Jan 11 '14 at 20:55
  • $\begingroup$ The most important issue is probably not the compassion of the atmosphere but whether it is possible to contain an atmosphere at all at any reasonable pressure. Using blankets to hold anything likely to retain atmospheric pressure would be a bigger challenge than making sure there wasn't too much CO and CO2. If you could, then all the other issues would be a lot less important. $\endgroup$ – matt_black Jan 12 '14 at 2:05
  • $\begingroup$ @matt_black: I disagree - maintaining pressure (especially when it's mostly oxygen and something like 0.15 atm might be sufficient) is a problem solvable through low-tech engineering, but the concerns I listed above could prove the method at the heart of the survival strategy fundamentally invalid. $\endgroup$ – Michael Borgwardt Jan 12 '14 at 18:09
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Summary

Okay, so after doing some serious number crunching and searching for literature values all over the web, I've come to the following conclusion: Yes, it is possible to survive in these conditions!

Plausibility of Environment

First, let us examine the plausibility of the environment which is described in the short story. Outside the safety of the living space, cordoned off by 30-or-so blankets, lined with aluminium foil, there is a vacuum, and all of the gases in the atmosphere have frozen out and precipitated onto Earth's surface. The exception is helium, which still is liquid. If we check the phase diagram of oxygen (note that in the figure below we would have to look at the pressure of 0.001), we see that indeed it is possible:

Phase diagram of oxygen

(Source)

Maintaining Pressure

The minimal pressure that humans need to survive (while breathing pure oxygen) is set at about 19 kPa, let's round that up to 20 kPa (0.2 atm). Let's furthermore also assume the space where this small family is living as about $3\times 3\times 2~\mathrm{m^3} = 18~\mathrm{m^3}$. Adding in the factor of a comfortable enough temperature (I pen that at $15~\mathrm{°C} = 288~\mathrm{K}$) we can use the ideal gas law: $$pV = nRT \Longleftrightarrow n=\frac{pV}{RT} = 150~\mathrm{mol\, \ce{O2}}$$

So we need $150~\mathrm{mol}$ of gas constantly in order to maintain pressurization at the desired temperature. Whether 30 heavy rugs really can do that I'll leave to an experimental physicist.

Oxygen Depletion

The fire isn't the only source of oxygen depletion, also the humans inhabiting the small space will use oxygen. Using this data, an average adult human being will consume $22.5~\mathrm{mol}$ $\ce{O2}$ per day. So we'll round this to $70~\mathrm{mol\, d^{-1}}$ for the whole family (2 adults, 2 kids). This is already corrected for oxygen exhalation, so the system generates an equal amount of $\ce{CO2}$.

Total Oxygen Balance and Required Heat

The only source of oxygen is the pail above the fire, in which the solid oxygen gets heated up to the boiling point (which, at 20 kPa is 77.1 K, see figure below; data source: NIST) and then further to ambient temperature. Density vs T of Oxygen at 0.2 kPa

Here some of the constants that I used for the following calculation: $c_\text{p} = 54~\mathrm{J\, mol^{-1}\, K^{-1}}$ for the liquid and $c_\text{p} = 29~\mathrm{J\, mol^{-1}\, K^{-1}}$ for the vapor phase with $\Delta H_\text{vap}(77.1~\mathrm{K},\; 20~\mathrm{kPa}) = 7.19~\mathrm{kJ\, mol^{-1}}$. The melting heat (at standard pressure, but let's just take this value from Wikipedia) is $0.445~\mathrm{kJ\, mol^{-1}}$, and the heat capacity of the solid I estimated at $25~\mathrm{J\, mol^{-1}\, K^{-1}}$. Initial temperature is $T_\text{i}=5~\mathrm{K}$, melting temperature (again for standard pressure, but we'll take it) $T_\text{m}=54~\mathrm{K}$, boiling point is given above and final temperature is $288~\mathrm{K}$.

The total heat needed for heating up $150~\mathrm{mol\, \ce{O2}}$ is $q_\text{tot, 150}=2.43~\mathrm{MJ}$.

Using the formula given in the paper Calculations of the Heat Release Rate by Oxygen Consumption for Various Applications I end up with $\dot{V} = 0.177~\mathrm{m^3\, d^{-1}}$, for a oxygen depletion coefficient of $0.8$ in a pure atmosphere of oxygen ($x_\mathrm{\ce{O2}}^0 = 1$, standard pressure is assumed in the formula). This is about 1% of the total volume in the cove, and is also only needed for the initial pressurization. However, since the inhabitants also produce $\ce{CO2}$ at non-negligible levels, and the concentration of said gas should stay under 2% for normal breathing and to avoid the feeling of dizziness, we need to up the oxygen generation rate.

So, to keep this whole system going with $x(\ce{CO2}) < 1.5~\%$, we need to generate 4700 mol $\ce{O2}$ every day (and this is without even looking at $\ce{CO2}$ generation by the fire).

It is inevitable that the exhaust gases from the fire should be led out via the chimney (that is also described in the story) without great mixing with the air in the room. But all in all the heat generation is possible (the oxygen needed to heat the 4700 mol to the temperature is only 46 mol), thus making it a survivable climate.

Conclusion

By the above estimation-based calculations and considering the limitations, I conclude that it would indeed be conceivable to survive in these conditions.

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  • $\begingroup$ It took me some time (I've been pondering the answer for so long, I decided to calculate the values just to be sure), but finally my answer is here! $\endgroup$ – tschoppi Feb 9 '14 at 13:46
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    $\begingroup$ Awesome answer! So if I understand correctly, one would need to generate much more oxygen than the humans need to survive, mainly to keep the CO2 they breathe out below toxic levels. But that's very good for the plausibility of the scenario since there's going to be a lot of leakage anyway. This leaves flammability as a main concern, as well as the amount of coal required - over 37kg each day, over 13 tons per year. But I think the least realistic aspect may be something else - the construction of a makeshift pressure suit to haul in the oxygen. $\endgroup$ – Michael Borgwardt Feb 9 '14 at 16:45
  • $\begingroup$ As to the leakage... $\ce{CO2}$ has a higher density ($0.37~\mathrm{kg\, m^{-3}}$ vs $0.27~\mathrm{kg\, m^{-3}}$ at 20 kPa) so it would be most beneficial to position the leak at the bottommost point in the room. $\endgroup$ – tschoppi Feb 9 '14 at 17:16
  • $\begingroup$ Some sort of intensive hydroponics would help, converting additional CO2 to O2. Presumably they need one anyway in order to produce food. $\endgroup$ – OrangeDog Nov 19 '18 at 11:17
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Yes, it is possible to produce a non-toxic atmosphere using a coal fire and frozen oxygen.

I'll answer your questions last to first.

No, low pressure oxygen will not make flammability worse, fires require something to bond with oxygen (often carbon, causing carbon dioxide). Low pressure will make things less flammable as there is less oxygen to bond with.

Yes, using an open coal fire in a semi-enclosed space will gain toxic levels of carbon dioxide, carbon monoxide and sulfur dioxide (among a few others) quickly. However with a minor modification from a open fire to a fire in a old fashioned chimney pipe stove (venting outside the hut) would eliminate this.

Lastly, Definitely Yes, a coal fire will produce more than enough energy to thaw the oxygen it uses. To quote figures 1 pound of coal will produce roughly 12,500 btu's (British Thermal Units) Assuming the temperature of the frozen oxygen we are thawing is -455°F (the "temperature of deep space") and about 3 degrees below the freezing temperature of helium. To thaw 1 gallon of the frozen oxygen will require roughly 900 btu's (if your stove has a 80% efficiency) and produce 860 gallons of pure oxygen at 80°F and atmospheric pressure. Coal will consume 140% of it's weight in pure oxygen, so for 1 pound of coal 135 gallons of your oxygen will be consumed. Leaving you with 11,500 btu's extra heat and 725 gallons of oxygen. The average Human consumes 25 gallons of oxygen per day making this plenty to survive on.

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  • $\begingroup$ That's just what I was looking for! Great to see that it would at least work in theory... $\endgroup$ – Michael Borgwardt Feb 7 '14 at 22:34
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No, it would not be possible to use a coal fire and frozen oxygen to create a atmosphere able to sustain life for any prolonged amount of time.

Liquid oxygen, though not explosive or flammable by itself, would mix with the soot given off by the coal fire and create a highly explosive mixture.

Also, a coal fire gives off a toxic amounts of carbon dioxide, and potentially other gases. Not to mention the soot, which would quickly suffocate anyone breathing it in.

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  • $\begingroup$ I think so too, but what I'm really looking for is concrete numbers concerning toxicity and energy budget. I see liquid oxygen and soot as a distraction from the core question, since ensuring that the fire produces a minimal amount of soot and that the oxygen is contained while it's liquid should not be too hard even with low-tech methods. $\endgroup$ – Michael Borgwardt Feb 6 '14 at 8:23
  • $\begingroup$ Sorry, I was thinking you were saying how they do it in the book. $\endgroup$ – Azzie Rogers Feb 6 '14 at 19:11

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