The danger is not only suffocation due to lack of oxygen, but also poisoning due to too much carbon dioxide in the air.
Normal air has 21% oxygen; humans will safely survive down to ~15%. Maybe 10% oxygen is barely survivable for a few hours. Mountaineers might have an advantage here, they regularly survive Everest, which has ⅓ of the oxygen at sea level available.
On the other hand, CO₂ is normally only present at very low concentration (0.04%).
Even 1% CO₂ in the air may lead to headaches and other health issues.
Occupational CO₂ exposure limits have been set in the United States at
0.5% (5000 ppm) for an eight-hour period. At this CO₂ concentration, International Space Station crew experienced headaches,
lethargy, mental slowness, emotional irritation, and sleep
disruption. Studies in animals at 0.5% CO₂ have demonstrated
kidney calcification and bone loss after eight weeks of exposure.
A study of humans exposed in 2.5 hour sessions demonstrated
significant effects on cognitive abilities at concentrations as low as
0.1% (1000ppm) CO₂ likely due to CO₂ induced increases in cerebral blood flow. Another study observed a decline in basic activity
level and information usage at 1000 ppm, when compared to 500
I am no expert on physiology, and I would not bet my life on the following calculation, but as a start:
If we assume that roughly one molecule of O₂ is turned into one molecule of CO₂, then each percent of O₂ gets turned into one percent of CO₂.
If we use 10% of the available oxygen, we will have reduced the oxygen concentration to 19% (which is no problem at all), but have 2% CO₂ (which is survivable for a few hours, but will already cause increased breathing and headaches).
This means that we can only use a small fraction of the available oxygen. If we ignored CO₂, we would think that we could survive down to maximally 10% O₂ (i.e.) use half of the available oxygen. While 10% O₂ mixed with 90% N₂ might be (barely) survivable, adding 11% CO₂ would lead to massive CO₂ poisoning.
Astronauts are budgeted 840g of O₂ per day. This corresponds to a volume of 1.1 m³.
From the calculation above, we can use only 10% of the available oxygen due to CO₂ poisoning, so we need ~10m³ oxygen or 50m³ air per day. This is on the safe side; probably astronauts are working hard most of the day.
A snow cave might be 2m×2m×2m = 8m³, which would give us a relatively safe (but uncomfortable) time between venting of about 4 hours.
Depending on the CO₂ tolerance, the person might get a bad headache after 1-2 hours, so this kind of room is too small to get a good night's sleep, especially for multiple people.
Even if we melt part of the snow and liberate the air trapped there, it can't be enough to sustain breathing; we would have to melt tens of cubic meters of snow per person per day.
This calculation assumes sea level pressures. At 5000m altitude, only about half of the oxygen is available.
The good news is that CO₂ will increase breathing and thus may warn us to get some fresh air. I am not sure that this is always reliable, though. Especially in a survival situation if already very tired.
From this calculation, it looks as if a snow cave needs a ventilation opening.
Addendum: Diffusion + Candles
Will CO₂ diffuse into the ice at such a rate that the occupation time of the shelter is extended?
CO₂ is the limiting factor, so no need to look at O₂ diffusion.
It is very difficult to calculate diffusion rates, especially into a heterogeneous system with a very irregular surface like ice.
Instead, I am looking at two related issues:
- How much water is needed to dissolve the CO₂?
- How fast can CO₂ permeate ice?
First I am looking a the equilibrium state, where as much CO₂ as possible has dissolved in the water / ice. I did not find any explicit values for solubility in ice, so I took the value for water at 0°C.
CO₂ dissolves very slowly in water (think of a soda maker), and probably even more slowly in ice, so the following calculation will
massively overestimate the amount of CO₂ that can realistically be absorbed. In 1, most of the experiments use bubbling and stirring for multiple
hours to equilibrate gas and liquid phases, while in our case the CO₂ would need to diffuse through at least a meter of ice. As we'll see in part 2, this will take forever.
Gas solubility follows Henry's law. The mole fraction of the gas in the liquid phase is proportional to the partial pressure in the gas phase.
The atmospheric pressure is 100 kPa. 2% CO₂ has a partial pressure of 2 kPa.
I will calculate how much water at 0°C we need to dissolve 1kg of CO₂ (amount exhaled per person per day) at a partial pressure of 2kPa.
Henry's constant varies a lot, but I have found values around 3e-2 mol/L/atm at room temperature. AT 0°C, CO₂ might be twice as soluble,
so I assume 6e-2. At 2% gas concentration, we can dissolve just 50mg in a liter of water.
To dissolve 1kg (output of one person / day), we need would 15 cubic meters of water. This is a lot. Even if we sat in a deep ice cave with a thick layer of ice around us, the CO₂ would need to diffuse through a meter of ice, which is not realistic:
I do not know the diffusion coefficient of CO₂ in snow, but it is very low in ice, according to this reference:
"the extremely small CO₂ diffusion coefficient in ice has not been accurately determined in the laboratory"
Another reference confirms this:
At -9.5°C the carbon dioxide permeation constant was found to be ...
about two million times less than in water. ... No permeation of
oxygen through the ice could be detected.
The first paper gives the permeation coefficient of CO₂ through ice as 1e-21 mol/(m s Pa).
If we assume that the walls of our cave are 1cm ice, with fresh air on the outside, then each square meter will let only a minuscule amount of CO₂ through:
1e-21 mol/(m s Pa) * 2 kPa * 1 m^2 / 0.01m * 3600s * 44g/mol = 3e-11 g/h
The paper mentions that CO₂ transport is not directly through the ice but along lattice defects with surface water, but I can't calculate that.
This leads some credibility to the argument that if an igloo glazes over on the inside the air will go bad. On the other hand, even if we replaced the igloo with a diving bell, the water wouldn't be able to absorb the CO₂ quickly enough. So I doubt that ice would be able to absorb any appreciable amount of CO₂.
We can also look at the issue from an anatomical side: The human lung has a surface area of around 50 square meters. On the body side the blood is circulated through the capillaries, on the air side the air is actively circulated as well.
If we put a 2nd barrier in series with the lung (our shelter), and require all of the CO₂ that passes through the lung to also pass through the shelter, then the shelter surface would have to be many times larger in surface area if we want to rely on natural convection and diffusion. As the shelter is much thicker than the capillary walls in the lung, the shelter surface would need to be astronomical.
So neither the solubility of CO₂, nor its diffusion coefficient in ice are large enough to remove much CO₂ from the air. We need to rely on ventilation.
1L. W. Diamond and N. N. Akinfiev, “Solubility of CO2 in water from −1.5 to 100 °C and from 0.1 to 100 MPa: evaluation of literature data and thermodynamic modelling,” Fluid Phase Equilibria, vol. 208, no. 1, pp. 265–290, Jun. 2003.
Candles as an oxygen detector:
A candle is a good test for enough oxygen (even though I would not bet my life on a candle):
15% oxygen will not support combustion of most flammable materials, but will still sustain life.
This was used in the Davy safety lamp for miners. It would not only detect methane, but also lack of oxygen.
These lamps were used for a long time and seemed to have been safe enough.
The effect of oxygen removal on a fire is shown in this video of fire and inert gas. A room full of people is flooded with an inert gas, the fire goes out, people stay alive. The gas is called Inergen, it is a mixture of nitrogen and argon and carbon dioxide; the latter increases the breathing rate and
makes it safe for humans. At least for a short time.
Without the carbon dioxide the human body would have no warning of a lack of oxygen.
I don't know if the candle would still be reliable at altitude.
I suspect that a fire depends mainly on the relative percentage of oxygen, not so much on the total pressure (as long as nothing dilutes the oxygen, it will burn even at reduced pressures).
On the other hand, humans need an absolute (partial) pressure of O₂.
While a candle might (probably? I have no evidence for this.) burn on the top of Mt. Everest, a human will slowly die from lack of oxygen. If that is true, a candle would not be a reliable indicator for enough oxygen on Mt. Everest.
More realistically, what will happen in an ice cave at 5000m altitude? In this case, we start with half of the oxygen at sea level.
This is already getting into dangerous territory for a non acclimated person. The candle flame won't mind, though.
If we consume 30% of the available oxygen, the candle might still be burning (at sea level, a candle will go out around 15% oxygen).
Now we have a situation of 15% oxygen at a low air pressure. I am not sure whether this would be a problem or not.
A candle will not detect toxic levels of carbon dioxide: as long as enough oxygen is present it will burn.
If there is a source of CO₂ (e.g. volcanic sources in Mt. Rainier ice caves),
a dangerous concentration of CO₂ might be present, with enough oxygen to support a candle.
If we replace 30% of the air with CO₂, we will have 15% oxygen and 30% carbon dioxide.
7-10% of carbon dioxide can be fatal. So a candle does not tell us the air is safe to breathe.
Normally people react strongly to CO₂ and will seek fresh air on their own, but I don't know whether this is still the case at altitude in a survival situation.
The candle will also use up some oxygen and produce more CO₂.
100g of candle will contain 85g of carbon. On burning, this candle will produce 311g of CO₂.
So a single candle produces CO₂ at a similar rate as a human and will increase the ventilation requirements accordingly.