4 ms·

D'oh! Just realized I was a chemical engineer an hour after I posted this :) The steps to solve this problem are pretty simple: 1) Find the diffusion coeffici

D'oh! Just realized I was a chemical engineer an hour after I posted this :) The steps to solve this problem are pretty simple: 1) Find the diffusion coefficient of oxygen in seawater. This can be found from data on ocean waters near the region the guy was trapped (Nigeria), modified to account for the cooler water under the surface. This link (http://www.unisense.com/files/PDF/Diverse/Seawater%20&%20Gases%20table.pdf http://www.unisense.com/files/PDF/Diverse/Seawater%20&%20Gas...) has these coefficients for various combinations of salinity and temperature. All of that data is for 1 bar so you'd have to modify those values for the pressure at ~30 m. 2) Set up an oxygen mass balance about a bubble of radius R. We'll assume the bubble is entirely spherical. If you consider the fact that the bubble is centered at a room corner, its radius will need to be sqrt(8) times bigger than the calculated value (since diffusion can only occur at the surface of the bubble that has contact with sea water). 3) The rate of change of O2 concentration (d_C_O2/dt) will be zero (in order for one to persist indefinitely in the bubble), and we know its value must be 15-19 vol % according to this site (http://www.newton.dep.anl.gov/askasci/zoo00/zoo00755.htm http://www.newton.dep.anl.gov/askasci/zoo00/zoo00755.htm). 4) Set up a O2 -> CO2 "reaction" (the lungs) just to get the molar ratios of each. This is the tricky part of the problem because this depends quite a bit on human physiology, and probably varies somewhat by the individual. 5) Plug these values into the radial form of Fick's law of diffusion, and solve the resulting equation for R. [If I get more time, I'll do this rigorously.]

marshray 13y ago
Is there any reason to think the result would be radically different than for blood? (E.g., hemoglobin). As the first comment points out, the lungs need about 70 m^2 of surface area. So as long as the surface area of the trapped bubble-seawater interface was at least that size, perhaps you could survive indefinitely, down to the limit of air-breathing depths. In smaller bubbles, perhaps you could keep yourself alive by splashing. Like an aquarium bubbler in reverse.
- Xcelerate 13y ago
  The problem is that the diffusion coefficient in the lungs is different than that for an air-water interface. Consider (as an extreme example) a bubble surrounded by steel with a surface area of 70 m^2. Yes, there is still diffusion of oxygen through the steel, but it's at a vastly lower rate than it is through water.
  - foobarbazqux 13y ago
    DC_water / DC_lungs * 70 m^2?
  - marshray 13y ago
    In case these help: https://en.wikipedia.org/wiki/Pulmonary_gas_pressures https://en.wikipedia.org/wiki/Pulmonary_gas_pressures > Following is a list of average partial pressures for a human at rest: > Lung Capillaries 20-40 pO2 (mmHg) > Alveolar air 35 pCO2 (Torr) http://www.engineeringtoolbox.com/oxygen-solubility-water-d_841.html http://www.engineeringtoolbox.com/oxygen-solubility-water-d_... Solubility of oxygen in equilibration with air in fresh and sea (salt) water - pressures ranging 1 - 4 bar abs http://publishing.cdlib.org/ucpressebooks/view?docId=kt167nb66r&chunk.id=d3_6_ch06&toc.id=ch06&brand=eschol http://publishing.cdlib.org/ucpressebooks/view?docId=kt167nb... In studies of the distribution of dissolved gases in the sea it is generally assumed that, whatever the location of a water particle, at some time it has been at the surface and in equilibrium with the air. In their studies of the dissolved nitrogen content Rakestraw and Emmel (1938a) have found that the water is virtually saturated (referred to a normal atmosphere), regardless of depth; therefore this assumption appears valid and also indicates that biological activity involving either fixation or production of nitrogen cannot be sufficient to affect significantly the concentration of this gas in the water. As the waters of the oceans appear to have been saturated with oxygen and carbon dioxide at some stage in their history when they were at the surface, the differences between the saturation values (computed from the temperatures and salinities) and the observed contents are measures of the changes which have been effected by biological agencies. The factors influencing the distribution of carbon dioxide are discussed in the following sections, and the distribution of dissolved oxygen will be considered in many places in the ensuing chapters.
tiatia 13y ago
I am not sure how much Fick's law has any influence. Most "mixing" inside and outside the bubble will be done by convection rather than by diffusion. Diffusion should have little to no influence on the whole problem. Assuming that you want to take into account mass transport through the phases (Air<->Water), you would have to apply Henry's Law.
- Xcelerate 13y ago
  Whoops. You're right. That's what I get for solving a mass transport problem at 5:00 AM.
carbocation 13y ago
Beautiful. For those playing at home, the answer to #4 depends mostly on diet, and for a typical Westerner it will most likely be 0.8. That is, slightly more O2 is used than CO2 is produced, representing a mixed metabolism of carbohydrates and fats.

marshray 13y ago

Is there any reason to think the result would be radically different than for blood? (E.g., hemoglobin). As the first comment points out, the lungs need about 70 m^2 of surface area. So as long as the surface area of the trapped bubble-seawater interface was at least that size, perhaps you could survive indefinitely, down to the limit of air-breathing depths. In smaller bubbles, perhaps you could keep yourself alive by splashing. Like an aquarium bubbler in reverse.

Xcelerate 13y ago
The problem is that the diffusion coefficient in the lungs is different than that for an air-water interface. Consider (as an extreme example) a bubble surrounded by steel with a surface area of 70 m^2. Yes, there is still diffusion of oxygen through the steel, but it's at a vastly lower rate than it is through water.
- foobarbazqux 13y ago
  DC_water / DC_lungs * 70 m^2?
- marshray 13y ago
  In case these help: https://en.wikipedia.org/wiki/Pulmonary_gas_pressures https://en.wikipedia.org/wiki/Pulmonary_gas_pressures > Following is a list of average partial pressures for a human at rest: > Lung Capillaries 20-40 pO2 (mmHg) > Alveolar air 35 pCO2 (Torr) http://www.engineeringtoolbox.com/oxygen-solubility-water-d_841.html http://www.engineeringtoolbox.com/oxygen-solubility-water-d_... Solubility of oxygen in equilibration with air in fresh and sea (salt) water - pressures ranging 1 - 4 bar abs http://publishing.cdlib.org/ucpressebooks/view?docId=kt167nb66r&chunk.id=d3_6_ch06&toc.id=ch06&brand=eschol http://publishing.cdlib.org/ucpressebooks/view?docId=kt167nb... In studies of the distribution of dissolved gases in the sea it is generally assumed that, whatever the location of a water particle, at some time it has been at the surface and in equilibrium with the air. In their studies of the dissolved nitrogen content Rakestraw and Emmel (1938a) have found that the water is virtually saturated (referred to a normal atmosphere), regardless of depth; therefore this assumption appears valid and also indicates that biological activity involving either fixation or production of nitrogen cannot be sufficient to affect significantly the concentration of this gas in the water. As the waters of the oceans appear to have been saturated with oxygen and carbon dioxide at some stage in their history when they were at the surface, the differences between the saturation values (computed from the temperatures and salinities) and the observed contents are measures of the changes which have been effected by biological agencies. The factors influencing the distribution of carbon dioxide are discussed in the following sections, and the distribution of dissolved oxygen will be considered in many places in the ensuing chapters.

tiatia 13y ago

I am not sure how much Fick's law has any influence. Most "mixing" inside and outside the bubble will be done by convection rather than by diffusion. Diffusion should have little to no influence on the whole problem. Assuming that you want to take into account mass transport through the phases (Air<->Water), you would have to apply Henry's Law.

Xcelerate 13y ago
Whoops. You're right. That's what I get for solving a mass transport problem at 5:00 AM.

carbocation 13y ago

Beautiful. For those playing at home, the answer to #4 depends mostly on diet, and for a typical Westerner it will most likely be 0.8. That is, slightly more O2 is used than CO2 is produced, representing a mixed metabolism of carbohydrates and fats.