## It Takes 150 Days to Suffocate if Sealed in a Room.

Greetings Estimateers! and welcome to today's exciting approximation.

We're getting real nerdy today, because our question is inspired by an episode of Doctor Who.  Specifically, series 3 episode 1:  Smith and Jones (in which the Doctor is in a hospital that is transported to the moon).

Here's the question:  You are trapped in a sealed room.  How long can you survive before you run out of air?

For this question, we need to estimate three things:
1. How much volume is in a typical room?
2. How quickly do people use up air?
3. When will people die due to lack of oxygen?

Let's start with the first question.  I'm in a room.  Its volume is approximately
$$(5$$ m$$)\times(15$$ m$$)\times(5$$ m$$) = 375$$ m$$^3 \approx 400$$ m$$^3 = 4\times 10^5$$ L

I will call this typical.

Question 2: How much air do I use up when I breathe?  According to HowStuffWorks a typical person breathes 8 L/min and 25% of that air is exhausted.  Thus, after one minute of breathing, our room goes from $$400000$$ L of air to $$399998$$ L of air.

We can use this number in two interpretations to get two different estimates of how long we can live in a sealed room.

ESTIMATION 1: Perfect breathing.

With "perfect breathing" we use exactly two liters every minute (the volume of air decreases linearly) and we die when all of the air is used up.  In this case we run out of air when
$$4 \times 10^5$$ L $$- 2 \frac{\text{L}}{\text{min}} T_{dead} = 0$$
$$T_{dead} = 2 \times 10^5$$ min $$\approx 150$$ days

By this estimation, it takes us 5 months to die if we are sealed in a room.

ESTIMATION 2: Imperfect breathing.

In this case, our breathing rate defines a decay time for usable air in the room.   Then, we die when we hit some critical percentage.  Assume (this is a raw guess) that we die when 3/4 of the oxygen is used up.  In this case, we will die when
$$4\times 10^5 e^{\ln(\frac{399998}{400000}) \times T_{dead}} = 1 \times 10^5$$.

Then,
$$T_{dead} \approx 2.8\times 10^5$$ min $$\approx 200$$ days.

Thus, with this estimation, we die after 200 days locked in a sealed room.

One thing worth noting about this estimate is that we seem to be living longer while breathing less efficiently.  While this seems paradoxical, it is being tacitly assumed in the second method that we could stay alive while using less oxygen than usual.  It is saying that there is some critical oxygen consumption rate that we need to survive.  We then use less and less per hour until, finally, we hit the critical rate and die.  But it is worth noting that our guesses are very close (order one corrections. Bah!).

So stop being so claustrophobic!  If you were sealed in an airtight room, it would take about half a year to suffocate.

1. I disagree with the result.
On face,it seems an an absurd figure.

Lets stick to estimation only.

your calculation only assumes that we live until all of oxygen is utilized. thats wrong !

There are two limiting case criteria here. Oxygen conc. and CO2 conc. safe levels. Am taking the extreme scenario...so its definitely death on the other side of these guidelines.

i did the math for both, turns out, we lose consciousness and suffer irreparable brain damage in 48 days 14hours and 40 min ( Oxygen levels below 14% : http://ehstoday.com/safety/confined-spaces/ehs_imp_32784/ )
with perfect breathing.

but if that was not good enough, CO2 levels rise above dangerous levels in just 20 days while to very dangerous levels in 34 days. however loss of consciousness is due to oxygen deficiency ( sad ) rather than with CO2 level ( 55 days time ).

needless to say, claustrophobia is a psychological issue with confinement.

2. In defense of the post, a few comments:

1) If you assume you die without using all the oxygen, you are not breathing perfectly. The "perfect breathing" assumption tacitly assumes you have some some of machine that you exhale into that throws away the "used up" air and keeps the "good air". Is this accurate if you are trapped in a room? Almost certainly not, but it is meant to give you an idea of what is possible if you used all the air as efficiently as you could. The reason, then, that imperfect breathing allows you to live longer is that it assumes you could stay alive while consuming less air over time.

2) Loss of consciousness and irreparable brain damage ain't dead. I checked your source and didn't see the 14% anywhere, but here's a source (http://www2.worksafebc.com/i/posters/2006/WS%2006_03.htm) that says you die at 10%. Repeating the imperfect breathing calculation with these numbers gives me 100 days, which is only a factor of 2 different than the original guess, which I totally do not care about (if we are always within a factor of 2 or 3, we are doing absurdly well).

Also, taking your number as a lower bound, giving us a range of about 50 to 150 days, still confirms the most interesting result of the estimate, which is that you should be much more worried about food and water than suffocation (this is not the case on many TV shows I've seen).

3. The link I posted clearly states the partial pressure at which there will not be any oxygen absorption in the lungs (60mm of Hg)...i think you can safely arrive at 14.45% oxygen concentration from there. More noteworthy is the fact that the Indian Standard Code for Ventilation lists 14% as the criteria for oxygen.

Here is the thing, this experiment is based on the premise of survival, which is assumed to be elastic...no matter how nauseous one feels or looses/gains weight...they survive if they live healthily after the trial without suffering significant permanent damage...atleast physically if not psychologically. I think you took the effects of CO2 increase very lightly. Combinind that with loss of unconsciousness due to Oxygen depravation ( and not CO2, but it will be dangerously high at 7% conc. then, only next to lethal )...any delay in the stopping of the trial for resuscitation will kill our subject. there is no survival then. and even if trial is stopped there is permanent brain damage to consider.

But all that aside,

Here is a notion to consider, "When do we die of lack of oxygen?" has a simple answer in the fact that it is its concentration. Your perfect breathing is a misleading term, but then again should have considered this basic science. I digress.

Again, in your imperfect breathing, you allow a person to be alive till oxygen concentration is 5% in the room.

Estimation doesn't mean you can ignore simple facts because you were too lazy to find or read them. But if you still believe so...you estimates misguide people and spread wrong data to an already misguided population.

You can simply correct your calculations above or you can defend them with "estimation", but if you are giving just 5 mins to type whatever your minds thinks up first about the problem and not actually getting even the basics right, i am sorry for you.

"but it is meant to give you an idea of what is possible if you used all the air as efficiently as you could. The reason, then, that imperfect breathing allows you to live longer is that it assumes you could stay alive while consuming less air over time."

1. it gives us an idea that is wrong by triple the actual "estimate".
2. "As efficiently as we could" means upto 60mm of Hg partial pressure...about 14% conc. then oxygen stops dissolving in blood...what more efficiency are you talking about?
3. We do not live longer with imperfect breathing. your numbers ( 3/4 consumption ) is way off the mark...it means 5% oxygen in room by volume...seriously.

I had read this post and liked it, but something about the numbers felt wrong. I looked up IS code 3103 & SP-32 ( that's the code of practice for ventilation, and handbook resp. ) and it listed 14%. Still i thought its just safety guideline and not lethal. However when all ASHRAE and OSHO codes listed oxygen safe at only upto 19.5%... that's when i started digging. Apparently Life and Brain are cheap commodities in India as far as Industrial practices go.
Here is an interesting fact, in a survey, it was found that openings and crevices never let Oxygen fall below 19%. So they altogether omitted any compulsory provisions for ventilation in the Factory Act 1984.

4. Let us forget the "perfect breathing" case.

In the "imperfect breathing" the limit used is that a person dies when the VOLUME consumed is 3/4. You seem to be saying something about 5% but I do not see any 5% in the said estimate. The volume of consumed oxygen is 75%.

The decay rate law then takes into account the fact that the lungs will tend to absorb in proportion to the amount present. Putting in the numbers gives the answer.

Instead of a 25% oxygen left if we use your number which is 14% then the answer will be $$3.8 \times 10^5$$ min which is about 263 days!

How are you getting 48 days anyway? You cannot use a linear approximation and then also use a threshold level of oxygen for death.

5. 60mm of Hg corresponds to 7.9%, which agrees more closely with my original guess than your estimate.

6. @jfb: one last time bear with the long comment, apologies in advance :)

Here goes...
Volume of the room = 400000 L = volume of air
Volume of oxygen = 84000L ( 21% )
Volume of CO2 = 120L ( ignored )

"perfect breathing" means efficiency of breathing, i.e amount of oxygen consumption per min is same for varying concentrations of oxygen. It does not mean we live until last of Oxygen is used up.

From second para under "The Respiratory System"
1. Normally oxygen intake in blood is at 110mm of Hg ( alveolar saturation )
2. 60mm of Hg being lethal point of no return, extrapolating it back, ( @jfb : you missed this ) we get that at 110mm of Hg partial pressure ( 60mm + 40mm due to CO2 in alveoli + 10mm due to vapor ) in the surroundings, the alveolar partial pressure will be 60mm.

Now to get 110mm of Hg partial pressure of oxygen in surroundings, the percentage of oxygen = 110/760 = 14.47%
which also very nicely corroborates with the criteria in the India code for ventilation. (jfb : see ?)

lets take 14% as lethal criteria then.

so when the oxygen volume in the room, falls from 84000 L to 56000 L ( 14% of 4E5 ), no dissolution in blood will take place, as 60mm of Hg partial pressure is reached in alveoli.

From HowStuffWorks link, per min we convert 0.4 lit of oxygen to carbon dioxide. which comes to 576 L per day.

The to bring down Oxygen to 56000L ( with constant rate "perfect breathing" )

(84000-56000)/576 = 48 days 14 hours and 40 min exact :)

Notice : I took "perfect breathing" to mean perfect breathing, as in upto the no-absorption limit, we keep getting oxygen @ 0.4 L per min. This is only for estimation sake.

However, I also had to check the CO2 criteria.

as you can now calculate, we produce 0.4 L of CO2 per min.
time to reach 3% by volume, i.e., 12000 L is about 20 days...and so forth for dangerous 5% (34 days ) and lethal 8% (55 days )....all values approx.

@Kartik : now coming to your assumption for imperfect breathing. My objection is towards this statement : "Assume (this is a raw guess) that we die when 3/4 of the oxygen is used up."

When 3/4 of oxygen is used up, what is the volume percentage of oxygen in the air ?
Ans = (0.25*84000)/4E5 = 5.25% which is clearly going over the top here. Even farther than your accepted 10%.

Apologies for irritating the hell out of jfb. hi ! nice to meet ya :)

@ bund : you know how nervous i get explaining math...let me know if any fatal error has been committed at my end.

7. @HellFragger

If you want to use a non-zero cutoff level like 14% say, then it is more correct to do the decay-rate type estimate ("imperfect breathing") than a linear one.

Suppose you do not agree with above statement. Let me ask you a question. Let us agree that 14% is the death level of oxygen. Now if you use this and do the calculation using both "perfect" and "imperfect" breathing methods you will get vastly different answers. Why do you think that happens?

8. Again, use whatever method you like, your premise for your calculations were plain wrong here.

You are now using "more correct", i do not care, just like the 80bn batteries.

the assumption in your calculations that we last untill all the oxygen is delpleted is wrong. and this time...there is a change by a factor of 10, moreoever its life and death.
But say you don't want to change your results, you can say you computed in months....then its just 2.6 months vs 5 months....hey...same order of magnitude !

i am done here !

9. Daniel I

1. Hell Fragger, you are clearly at the wrong place. You want wedocalculationsaccuratetoonehudreddecimalplaces.blogspot.com

2. Interesting note, in the case of a volatile chemical spill in your neighboorhood, the recommended emergency procedure is to go into your room with the least windows and put plastic over the windows and the doorway. This post made me feel better about that.

10. @Daniel I : by using Reductio ad absurdum...you have shown what you are, and have clearly missed the whole point of discussion. Internet is a wonderful democratic place. Thrive...
( if you really are curious, no you comment wasn't as witty as you thought it was )

FYI : the discussion here, wasn't about accuracy, rather about the basic assumptions of survival which are wrong.

11. anyhow, now i am curious, as to where i made error in calculation.

lets start again,

Estimation 1 : "we use exactly 2 liters every min ". you mean "Air" of course ! you arrived at this with the assumption, as long as oxygen remains we can breathe/use it.
From the same link ( Howstuffworks ), we exhale out air that is still 15% oxygen.
We chemically absorb 0.4 lit per min. If we go by your assumption, we lock our subject in a evacuated room, and introduce 2 litre of "Air" ( yes, it has 0.4 lit of oxygen in it ), our subject should be able to live for exactly 1 min ?

i think not ! as i pointed out above.

estimation is fine, i get it. What i don't "get" is wrong premises/assumptions in estimation.

don't be exact with calculation, but give enough thought to survival of our subject. Else its misguidance online !

12. Anonymous

Just so we have some quorum here, I don't know what HellFrager is like in real life, but as far as I can tell from his posts, he is acts like an ass, sounds like an ass, and seems to think the way to arguments is to plug his ears with his hands and start shouting as loud as he can.

You know, we understand what you're saying, give the readers some respect. Some ppl will agree to what you say, others wont, make your point and move on, its not about democracy or freedom of speech really, just about common sense, if your argument is so strong, you wouldnt have to be shouting it in ten different ways.

damn, got my knickers in a twist for no reason

13. Anonymous

I am trying to put in place a Plan for a "Chemical or Germ Warfare". I read in such a case, it is best to "Shelter-In-Place" by sealing off a room to get through such a disaster. My concern for my family is, how long could we sustain in a sealed off, Shelter-In-Place before suffication poses an additional threat?